OPEN - $500

If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then
\[N \gg 2^{n}.\]

Erdős called this 'perhaps my first serious problem'. The powers of $2$ show that $2^n$ would be best possible here. The trivial lower bound is $N \gg 2^{n}/n$, since all $2^n$ distinct subset sums must lie in $[0,Nn)$. Erdős and Moser [Er56] proved
\[ N\geq (\tfrac{1}{4}-o(1))\frac{2^n}{\sqrt{n}}.\]
(In [Er85c] Erdős offered \$100 for any improvement of the constant $1/4$ here.)

A number of improvements of the constant have been given (see [St23] for a history), with the current record $\sqrt{2/\pi}$ first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu [DFX21], who in fact prove the exact bound $N\geq \binom{n}{\lfloor n/2\rfloor}$.

In [Er73] and [ErGr80] the generalisation where $A\subseteq (0,N]$ is a set of real numbers such that the subset sums all differ by at least $1$ is proposed, with the same conjectured bound. (The second proof of [DFX21] applies also to this generalisation.)

This problem appears in Erdős' book with Spencer [ErSp74] in the final chapter titled 'The kitchen sink'. As Ruzsa writes in [Ru99] "it is a rich kitchen where such things go to the sink".

The sequence of minimal $N$ for a given $n$ is A276661 in the OEIS.

See also [350].

SOLVED - $1000

Can the smallest modulus of a covering system be arbitrarily large?

Described by Erdős as 'perhaps my favourite problem'. Hough [Ho15], building on work of Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], has shown (contrary to Erdős' expectations) that the answer is no: the smallest modulus must be at most $10^{18}$.

An alternative, simpler, proof was given by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22], who improved the bound on the smallest modulus to $616000$.

SOLVED - $10000

For any $C>0$ are there infinitely many $n$ such that
\[p_{n+1}-p_n> C\frac{\log\log n\log\log\log\log n}{(\log\log \log n)^2}\log n?\]

The peculiar quantitative form of Erdős' question was motivated by an old result of Rankin [Ra38], who proved there exists some constant $C>0$ such that the claim holds. Solved by Maynard [Ma16] and Ford, Green, Konyagin, and Tao [FGKT16]. The best bound available, due to all five authors [FGKMT18], is that there are infinitely many $n$ such that
\[p_{n+1}-p_n\gg \frac{\log\log n\log\log\log\log n}{\log\log \log n}\log n.\]
The likely truth is a lower bound like $\gg(\log n)^2$. In [Er97c] Erdős revised the value of this problem to \$5000 and reserved the \$10000 for a lower bound of $>(\log n)^{1+c}$ for some $c>0$.

See also [687].

OPEN

Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that
\[\lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C?\]

Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\log n$. This problem asks whether $S=[0,\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:

- $\infty\in S$ by Westzynthius' result [We31] on large prime gaps,
- $0\in S$ by the work of Goldston, Pintz, and Yildirim [GPY09] on small prime gaps,
- Erdős [Er55] and Ricci [Ri56] independently showed that $S$ has positive Lebesgue measure,
- Hildebrand and Maier [HiMa88] showed that $S$ contains arbitrarily large (finite) numbers,
- Pintz [Pi16] showed that there exists some small constant $c>0$ such that $[0,c]\subset S$,
- Banks, Freiberg, and Maynard [BFM16] showed that at least $12.5\%$ of $[0,\infty)$ belongs to $S$,
- Merikoski [Me20] showed that at least $1/3$ of $[0,\infty)$ belongs to $S$, and that $S$ has bounded gaps.

SOLVED - $100

Let $d_n=p_{n+1}-p_n$. Are there infinitely many $n$ such that $d_n<d_{n+1}<d_{n+2}$?

Conjectured by Erdős and Turán [ErTu48]. Shockingly Erdős offered \$25000 for a disproof of this, but as he comments, it 'is certainly true'. (In [Er85c] he goes further and offers 'all the money I can earn, beg, borrow or steal for [a disproof]'.)

Indeed, the answer is yes, as proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]. They in fact prove that, for any $m\geq 1$, there are infinitely many $n$ such that \[d_n<d_{n+1}<\cdots <d_{n+m}\] and infinitely many $n$ such that \[d_n> d_{n+1}>\cdots >d_{n+m}.\]

OPEN

Is there a covering system all of whose moduli are odd?

Asked by Erdős and Selfridge (sometimes also with Schinzel). They also asked whether there can be a covering system such that all the moduli are odd and squarefree. The answer to this stronger question is no, proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].

Hough and Nielsen [HoNi19] proved that at least one modulus must be divisible by either $2$ or $3$. A simpler proof of this fact was provided by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].

Selfridge has shown (as reported in [Sc67]) that such a covering system exists if a covering system exists with moduli $n_1,\ldots,n_k$ such that no $n_i$ divides any other $n_j$ (but the latter has been shown not to exist, see [586]).

OPEN - $500

If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$.

Conjectured by Erdős and Turán. They also suggest the stronger conjecture that $\limsup 1_A\ast 1_A(n)/\log n>0$.

Another stronger conjecture would be that the hypothesis $\lvert A\cap [1,N]\rvert \gg N^{1/2}$ for all large $N$ suffices.

Erdős and Sárközy conjectured the stronger version that if $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ with $a_n/b_n\to 1$ are such that $A+B=\mathbb{N}$ then $\limsup 1_A\ast 1_B(n)=\infty$.

See also [40].

OPEN - $1000

Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$,
\[h(N) = N^{1/2}+O_\epsilon(N^\epsilon)?\]

A problem of Erdős and Turán. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erdős remarks this is perhaps too optimistic. Erdős and Turán [ErTu41] proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindström [Li69]. Both proofs in fact give
\[h(N) \leq N^{1/2}+N^{1/4}+1.\]
Balogh, Füredi, and Roy [BFR21] improved the bound in the error term to $0.998N^{1/4}$, which has been further optimised by O'Bryant [OB22] to yield
\[h(N)\leq N^{1/2}+0.99703N^{1/4}\]
for sufficiently large $N$.

OPEN

Is there a set $A\subset\mathbb{N}$ such that
\[\lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2)\]
and such that every large integer can be written as $p+a$ for some prime $p$ and $a\in A$?

Can the bound $O(\log N)$ be achieved? Must such an $A$ satisfy \[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N}> 1?\]

Such a set is called an additive complement to the primes.

Erdős [Er54] proved that such a set $A$ exists with $\lvert A\cap\{1,\ldots,N\}\rvert\ll (\log N)^2$ (improving a previous result of Lorentz [Lo54] who achieved $\ll (\log N)^3$). Wolke [Wo96] has shown that such a bound is almost true, in that we can achieve $\ll (\log N)^{1+o(1)}$ if we only ask for almost all integers to be representable.

The answer to the third question is yes: Ruzsa [Ru98c] has shown that we must have \[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N}\geq e^\gamma\approx 1.781.\]

OPEN

Find the optimal constant $c>0$ such that the following holds.

For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partition into two equal parts, so that $\lvert A\rvert=\lvert B\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\in A$ and $b\in B$ is at least $cN$.

The minimum overlap problem. The example (with $N$ even) $A=\{N/2+1,\ldots,3N/2\}$ shows that $c\leq 1/2$ (indeed, Erdős initially conjectured that $c=1/2$). The lower bound of $c\geq 1/4$ is trivial, and Scherk improved this to $1-1/\sqrt{2}=0.29\cdots$. The current records are
\[0.379005 < c < 0.380926\cdots,\]
the lower bound due to White [Wh22] and the upper bound due to Haugland [Ha16].

SOLVED

We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0<d_s(B)<1$ where $d_s$ is the Schnirelmann density.

Can a lacunary set $A\subset\mathbb{N}$ be an essential component?

The answer is no by Ruzsa [Ru87], who proved that if $A$ is an essential component then there exists some constant $c>0$ such that $\lvert A\cap \{1,\ldots,N\}\rvert \geq (\log N)^{1+c}$ for all large $N$.

OPEN - $500

Is there an infinite Sidon set $A\subset \mathbb{N}$ such that
\[\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}\]
for all $\epsilon>0$?

The trivial greedy construction achieves $\gg N^{1/3}$. The current best bound of $\gg N^{\sqrt{2}-1+o(1)}$ is due to Ruzsa [Ru98]. (Erdős [Er73] had offered \$25 for any construction which achieves $N^{c}$ for some $c>1/3$.) Erdős proved that for every infinite Sidon set $A$ we have
\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0,\]
and also that there is a set $A\subset \mathbb{N}$ with $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ such that $1_A\ast 1_A(n)=O(1)$.

Erdős and Rényi have constructed, for any $\epsilon>0$, a set $A$ such that \[\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}\] for all large $N$ and $1_A\ast 1_A(n)\ll_\epsilon 1$ for all $n$.

SOLVED - $500

If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist $d,m\geq 1$ such that
\[\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?\]

The Erdős discrepancy problem. This is true, and was proved by Tao [Ta16], who also proved the more general case when $f$ takes values on the unit sphere.

In [Er81] it is further conjectured that \[\max_{md\leq x}\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert \gg \log x.\]

In [Er85c] Erdős also asks about the special case when $f$ is multiplicative.

OPEN - $250

Find the value of $\lim_{k\to \infty}R(k)^{1/k}$.

Erdős offered \$100 for just a proof of the existence of this constant, without determining its value. He also offered \$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. Erdős proved
\[\sqrt{2}\leq \liminf_{k\to \infty}R(k)^{1/k}\leq \limsup_{k\to \infty}R(k)^{1/k}\leq 4.\]
The upper bound has been improved to $4-\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe [CGMS23].

This problem is #3 in Ramsey Theory in the graphs problem collection.

OPEN - $500

Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances?

A $\sqrt{n}\times\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz [GuKa15] who proved that there are always $\gg n/\log n$ many distinct distances.

A stronger form (see [604]) may be true: is there a single point which determines $\gg n/\sqrt{\log n}$ distinct distances, or even $\gg n$ many such points, or even that this is true averaged over all points.

See also [661].

OPEN - $500

Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(1/\log\log n)}$ many pairs which are distance 1 apart?

The unit distance problem. In [Er94b] Erdős dates this conjecture to 1946.

This would be the best possible, as is shown by a set of lattice points. It is easy to show that there are $O(n^{3/2})$ many such pairs. The best known upper bound is $O(n^{4/3})$, due to Spencer, Szemerédi, and Trotter [SST84]. In [Er83c] and [Er85] Erdős offers \$250 for an upper bound of the form $n^{1+o(1)}$.

Part of the difficulty of this problem is explained by a result of Valtr (see [Sz16]), who constructed a metric on $\mathbb{R}^2$ and a set of $n$ points with $\gg n^{4/3}$ unit distance pairs (with respect to this metric). The methods of the upper bound proof of Spencer, Szemerédi, and Trotter [SST84] generalise to include this metric. Therefore to prove an upper bound better than $n^{4/3}$ some special feature of the Euclidean metric must be exploited.

See a survey by Szemerédi [Sz16] for further background and related results.

SOLVED

If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor \frac{n+1}{2}\rfloor$ distinct distances.

Solved by Altman [Al63]. The stronger variant that says there is one point which determines at least $\lfloor \frac{n+1}{2}\rfloor$ distinct distances is still open. Fishburn in fact conjectures that if $R(x)$ counts the number of distinct distances from $x$ then
\[\sum_{x\in A}R(x) \geq \binom{n}{2}.\]

Szemerédi conjectured (see [Er97e]) that this stronger variant remains true if we only assume that no three points are on a line, and proved this with the weaker bound of $n/3$.

See also [660].

OPEN - $100

Does every convex polygon have a vertex with no other $4$ vertices equidistant from it?

Erdős originally conjectured this with no $3$ vertices equidistant, but Danzer found a convex polygon on 9 points such that every vertex has three vertices equidistant from it (but this distance depends on the vertex), and Fishburn and Reeds [FiRe92] have found a convex polygon on 20 points such that every vertex has three vertices equidistant from it (and this distance is the same for all vertices).

If this fails for $4$, perhaps there is some constant for which it holds?

Erdős suggested this as an approach to solve [96]. Indeed, if this problem holds for $k+1$ vertices then, by induction, this implies an upper bound of $kn$ for [96].

The answer is no if we omit the requirement that the polygon is convex (I thank Boris Alexeev and Dustin Mixon for pointing this out), since for any $d$ there are graphs with minimum degree $d$ which can be embedded in the plane such that each edge has length one (for example one can take the $d$-dimensional hypercube graph on $2^d$ vertices). One can then connect the vertices in a cyclic order so that there are no self-intersections and no three consecutive vertices on a line, thus forming a (non-convex) polygon.

OPEN - $500

Let $f(n)$ be minimal such that any $f(n)$ points in $\mathbb{R}^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n)=2^{n-2}+1$.

The Erdős-Klein-Szekeres 'Happy Ending' problem. The problem originated in 1931 when Klein observed that $f(4)=5$. Turán and Makai showed $f(5)=9$. Erdős and Szekeres proved the bounds
\[2^{n-2}+1\leq f(n)\leq \binom{2n-4}{n-2}+1.\]
([ErSz60] and [ErSz35] respectively). There were several improvements of the upper bound, but all of the form $4^{(1+o(1))n}$, until Suk [Su17] proved
\[f(n) \leq 2^{(1+o(1))n}.\]
The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20], who prove
\[f(n) \leq 2^{n+O(\sqrt{n\log n})}.\]

In [Er97e] Erdős clarifies that the \$500 is for a proof, and only offers \$100 for a disproof.

This problem is #1 in Ramsey Theory in the graphs problem collection.

SOLVED

If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true that
\[\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?\]

The lower bound is easy: this is $\geq n$ and equality holds if and only if $p(z)=z^n$. The assumption that the set is connected is necessary, as witnessed for example by $p(z)=z^2+10z+1$.

The Chebyshev polynomials show that $n^2/2$ is best possible here. Erdős originally conjectured this without the $o(1)$ term but Szabados observed that was too strong. Pommerenke [Po59a] proved an upper bound of $\frac{e}{2}n^2$.

Eremenko and Lempert [ErLe94] have shown this is true, and in fact Chebyshev polynomials are the extreme examples.

SOLVED

Let $p(z)=\prod_{i=1}^n (z-z_i)$ for $\lvert z_i\rvert \leq 1$. Then the area of the set where \[A=\{ z: \lvert p(z)\rvert <1\}\]
is $>n^{-O(1)}$ (or perhaps even $>(\log n)^{-O(1)}$).

Conjectured by Erdős, Herzog, and Piranian [ErHePi58]. The lower bound $\mu(A) \gg n^{-4}$ follows from a result of Pommerenke [Po61]. The stronger lower bound $\gg (\log n)^{-O(1)}$ is still open.

Wagner [Wa88] proves, for $n\geq 3$, the existence of such polynomials with \[\mu(A) \ll_\epsilon (\log\log n)^{-1/2+\epsilon}\] for all $\epsilon>0$.

SOLVED - $100

Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$, and for $n\geq 1$ let
\[p_n(z)=\prod_{i\leq n} (z-z_i).\]
Let $M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert$. Is it true that $\limsup M_n=\infty$? Is it true that there exists $c>0$ such that for infinitely many $n$ we have $M_n > n^c$, or even that for all $n$
\[\sum_{k\leq n}M_k > n^{1+c}?\]

The weaker conjecture that $\limsup M_n=\infty$ was proved by Wagner, who show that there is some $c>0$ with $M_n>(\log n)^c$ infinitely often.

This was solved by Beck [Be91], who proved that there exists some $c>0$ such that \[\max_{n\leq N} M_n > N^c.\]

OPEN

Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\to \infty$.

SOLVED - $1000

Let $r_k(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain a non-trivial $k$-term arithmetic progression. Prove that $r_k(N)=o(N)$.

Proved by Szemerédi [Sz74]. The best known bounds are due to Kelley and Meka [KeMe23] for $k=3$ (with further slight improvements in [BlSi23]), Green and Tao [GrTa17] for $k=4$, and Leng, Sah, and Sawhney [LSS24] for $k\geq 5$.

See also [3].

OPEN - $500

Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k\geq 1$ we have
\[ \lvert kx -y\rvert \geq 1.\]
Does this imply that
\[\liminf \frac{\lvert A\cap [1,x]\rvert}{x}=0?\]
Or
\[\sum_{x\in A}\frac{1}{x\log x}<\infty,\]
or
\[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)?\]
Perhaps even
\[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}\ll \frac{\log x}{\sqrt{\log\log x}}?\]

Note that if $A$ is a set of integers then the condition implies that $A$ is a primitive set (that is, no element of $A$ is divisible by any other), for which the convergence of $\sum_{n\in A}\frac{1}{n\log n}$ was proved by Erdős [Er35], and the upper bound
\[\sum_{n<x}\frac{1}{n}\ll \frac{\log x}{\sqrt{\log\log x}}\]
was proved by Behrend [Be35].

In [Er73] mentions an unpublished proof of Haight that \[\lim \frac{\lvert A\cap [1,x]\rvert}{x}=0\] holds if the elements of $A$ are independent over $\mathbb{Q}$.

See also [858].

SOLVED - $250

The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is equal to $1$.

It is known that there exists some constant $c>0$ such that for large $k$
\[c\frac{k^2}{\log k}\leq R(3,k) \leq (1+o(1))\frac{k^2}{\log k}.\]
The lower bound is due to Kim [Ki95], the upper bound is due to Shearer [Sh83], improving an earlier bound of Ajtai, Komlós, and Szemerédi [AjKoSz80]. The lower bound has been improved to
\[\left(\frac{1}{4}-o(1)\right)\frac{k^2}{\log k}\]
independently by Bohman and Keevash [BoKe21] and Pontiveros, Griffiths and Morris [PGM20]. The latter collection of authors conjecture that this lower bound is the true order of magnitude.

See also [544].

OPEN - $250

Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determine
\[\lim_{k\to \infty}R(3;k)^{1/k}.\]

Erdős offers \$100 for showing that this limit is finite. An easy pigeonhole argument shows that
\[R(3;k)\leq 2+k(R(3;k-1)-1),\]
from which $R(3;k)\leq \lceil e k!\rceil$ immediately follows. The best-known upper bounds are all of the form $ck!+O(1)$, and arise from this type of inductive relationship and computational bounds for $R(3;k)$ for small $k$. The best-known lower bound (coming from lower bounds for Schur numbers) is due to Exoo [Ex94],
\[R(3;k) \gg (321)^{k/5}.\]

See also [483].

OPEN

Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\equiv a_i\pmod{n_i}$ for at most one $1\leq i\leq r$). How large can $r$ be in terms of $N$?

Let $f(N)$ be the maximum possible $r$. Erdős and Stein conjectured that $f(N)=o(N)$, which was proved by Erdős and Szemerédi [ErSz68], who showed that, for every $\epsilon>0$,
\[\frac{N}{\exp((\log N)^{1/2+\epsilon})} \ll_\epsilon f(N) < \frac{N}{(\log N)^c}\]
for some $c>0$. Erdős believed the lower bound is closer to the truth.

OPEN

Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\epsilon>0$ and large $n$,
\[s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}?\]
Is it true that
\[s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}?\]

Erdős [Er51] showed that there are infinitely many $n$ such that
\[s_{n+1}-s_n > (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n},\]
so this bound would be the best possible.

In [Er79] Erdős says perhaps $s_{n+1}-s_n \ll \log s_n$, but he is 'very doubtful'.

Filaseta and Trifonov [FiTr92] proved an upper bound of $s_n^{1/5}$. Pandey [Pa24] has improved this exponent to $1/5-c$ for some constant $c>0$.

SOLVED

Let $f(n)$ be minimal such that the following holds. For any $n$ points in $\mathbb{R}^2$, not all on a line, there must be at least $f(n)$ many lines which contain exactly 2 points (called 'ordinary lines'). Does $f(n)\to \infty$? How fast?

Conjectured by Erdős and de Bruijn. The Sylvester-Gallai theorem states that $f(n)\geq 1$. The fact that $f(n)\geq 1$ was conjectured by Sylvester in 1893. Erdős rediscovered this conjecture in 1933 and told it to Gallai who proved it.

That $f(n)\to \infty$ was proved by Motzkin [Mo51]. Kelly and Moser [KeMo58] proved that $f(n)\geq\tfrac{3}{7}n$ for all $n$. This is best possible for $n=7$. Motzkin conjectured that for $n\geq 13$ there are at least $n/2$ such lines. Csima and Sawyer [CsSa93] proved a lower bound of $f(n)\geq \tfrac{6}{13}n$ when $n\geq 8$. Green and Tao [GrTa13] proved that $f(n)\geq n/2$ for sufficiently large $n$. (A proof that $f(n)\geq n/2$ for large $n$ was earlier claimed by Hansen but this proof was flawed.)

The bound of $n/2$ is best possible for even $n$, since one could take $n/2$ points on a circle and $n/2$ points at infinity. Surprisingly, Green and Tao [GrTa13] show that if $n$ is odd then $f(n)\geq 3\lfloor n/4\rfloor$.

OPEN

Is there a dense subset of $\mathbb{R}^2$ such that all pairwise distances are rational?

Conjectured by Ulam. Erdős believed there cannot be such a set. This problem is discussed in a blogpost by Terence Tao, in which he shows that there cannot be such a set, assuming the Bombieri-Lang conjecture. The same conclusion was independently obtained by Shaffaf [Sh18].

Indeed, Shaffaf and Tao actually proved that such a rational distance set must be contained in a finite union of real algebraic curves. Solymosi and de Zeeuw [SdZ10] then proved (unconditionally) that a rational distance set contained in a real algebraic curve must be finite, unless the curve contains a line or a circle.

Ascher, Braune, and Turchet [ABT20] observed that, combined, these facts imply that a rational distance set in general position must be finite (conditional on the Bombieri-Lang conjecture).

OPEN

Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.

In [Er85c] Erdős also conjectures that $d_n=d_{n+1}=\cdots=d_{n+k}$ is solvable for every $k$.

SOLVED - $500

Let $n\geq 1$ and
\[A=\{a_1<\cdots <a_{\phi(n)}\}=\{ 1\leq m<n : (m,n)=1\}.\]
Is it true that
\[ \sum_{1\leq k<\phi(n)}(a_{k+1}-a_k)^2 \ll \frac{n^2}{\phi(n)}?\]

SOLVED

Is there a set $A\subset\mathbb{N}$ such that, for all large $N$,
\[\lvert A\cap\{1,\ldots,N\}\rvert \ll N/\log N\]
and such that every large integer can be written as $2^k+a$ for some $k\geq 0$ and $a\in A$?

Lorentz [Lo54] proved there is such a set with, for all large $N$,
\[\lvert A\cap\{1,\ldots,N\}\rvert \ll \frac{\log\log N}{\log N}N\]
The answer is yes, proved by Ruzsa [Ru72]. Ruzsa's construction is ingeniously simple:
\[A = \{ 5^nm : m\geq 1\textrm{ and }5^n\geq C\log m\}+\{0,1\}\]
for some large absolute constant $C>0$. That every large integer is of the form $2^k+a$ for some $a\in A$ is a consequence of the fact that $2$ is a primitive root of $5^n$ for all $n\geq 1$.

In [Ru01] Ruzsa constructs an asymptotically best possible answer to this question (a so-called 'exact additive complement'); that is, there is such a set $A$ with \[\lvert A\cap\{1,\ldots,N\}\rvert \sim \frac{N}{\log_2N}\] as $N\to \infty$.

OPEN

Let $n_1<n_2<\cdots$ be the sequence of integers which are the sum of two squares. Explore the behaviour of (i.e. find good upper and lower bounds for) the consecutive differences $n_{k+1}-n_k$.

Erdős [Er51] proved that, for infinitely many $k$,
\[ n_{k+1}-n_k \gg \frac{\log n_k}{\sqrt{\log\log n_k}}.\]
Richards [Ri82] improved this to
\[\limsup_{k\to \infty} \frac{n_{k+1}-n_k}{\log n_k} \geq 1/4.\]
The constant $1/4$ here has been improved, most lately to $0.868\cdots$ by Dietmann, Elsholtz, Kalmynin, Konyagin, and Maynard [DEKKM22].
The best known upper bound is due to Bambah and Chowla [BaCh47], who proved that
\[n_{k+1}-n_k \ll n_k^{1/4}.\]

The differences are listed at A256435 on the OEIS.

SOLVED

If $A\subseteq \mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determine an obtuse angle.

For $d=2$ this is trivial. For $d=3$ there is an unpublished proof by Kuiper and Boerdijk. The general case was proved by Danzer and Grünbaum [DaGr62].

SOLVED

Let
\[ f(\theta) = \sum_{k\geq 1}c_k e^{ik\theta}\]
be a trigonometric polynomial (so that the $c_k\in \mathbb{C}$ are finitely supported) with real roots such that $\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1$. Then
\[\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.\]

This was solved independently by Kristiansen [Kr74] (only in the case when $c_k\in\mathbb{R}$) and Saff and Sheil-Small [SSS73] (for general $c_k\in \mathbb{C}$).

SOLVED

Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function. Is it true that if
\[\lim_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}\]
exists then it must be $0$?

SOLVED

Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $\pm 1$, such that
\[\sqrt{n} \ll \lvert P(z) \rvert \ll \sqrt{n}\]
for all $\lvert z\rvert =1$, with the implied constants independent of $z$ and $n$?

Originally a conjecture of Littlewood. The answer is yes (for all $n\geq 2$), proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].

See also [230].

SOLVED

Let $(S_n)_{n\geq 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire function $f(z)$ such that, for all $n\geq 1$, there exists some $k_n\geq 0$ such that
\[f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?\]

Solved in the affirmative by Barth and Schneider [BaSc72].

SOLVED

Let $P(z)=\sum_{1\leq k\leq n}a_kz^k$ for some $a_k\in \mathbb{C}$ with $\lvert a_k\rvert=1$ for $1\leq k\leq n$. Does there exist a constant $c>0$ such that, for $n\geq 2$, we have
\[\max_{\lvert z\rvert=1}\lvert P(z)\rvert \geq (1+c)\sqrt{n}?\]

The lower bound of $\sqrt{n}$ is trivial from Parseval's theorem. The answer is no (contrary to Erdős' initial guess). Kahane [Ka80] constructed 'ultraflat' polynomials $P(z)=\sum a_kz^k$ with $\lvert a_k\rvert=1$ such that
\[P(z)=(1+o(1))\sqrt{n}\]
uniformly for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$, where the $o(1)$ term $\to 0$ as $n\to \infty$.

For more details see the paper [BoBo09] of Bombieri and Bourgain and where Kahane's construction is improved to yield such a polynomial with \[P(z)=\sqrt{n}+O(n^{\frac{7}{18}}(\log n)^{O(1)})\] for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$.

See also [228].

SOLVED

Let $S$ be a string of length $2^k-1$ formed from an alphabet of $k$ characters. Must $S$ contain an abelian square: two consecutive blocks $x$ and $y$ such that $y$ is a permutation of $x$?

Erdős initially conjectured that the answer is yes for all $k\geq 2$, but for $k=4$ this was disproved by de Bruijn and Erdős. At least, this is what Erdős writes, but gives no construction or reference, and a simple computer search produces no such counterexamples for $k=4$. Perhaps Erdős meant $2^k$, where indeed there is an example for $k=4$:
\[1213121412132124.\]

Erdős then asked if there is in fact an infinite string formed from $\{1,2,3,4\}$ which contains no abelian squares? This is equivalent to [192], and such a string was constructed by Keränen [Ke92]. The existence of this infinite string gives a negative answer to the problem for all $k\geq 4$.

Containing no abelian squares is a stronger property than being squarefree (the existence of infinitely long squarefree strings over alphabets with $k\geq 3$ characters was established by Thue).

We refer to a recent survey by Fici and Puzynina [FiPu23] for more background and related results.

OPEN

Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Is it true that $f(n)=o(\log n)$?

Erdős [Er50] proved that there are infinitely many $n$ such that $f(n)\gg \log\log n$. Erdős could not even prove that there do not exist infinitely many integers $n$ such that for all $2^k<n$ the number $n-2^k$ is prime (probably $n=105$ is the largest such integer).

The sequence of values of $f(n)$ is A109925 on the OEIS.

See also [237].

SOLVED

Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N$ for all large $N$. Let $f(n)$ count the number of solutions to $n=p+a$ for $p$ prime and $a\in A$. Is it true that $\limsup f(n)=\infty$?

SOLVED

Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function. Is it true that
\[ \lim_{N\to \infty}\frac{1}{N}\sum_{n\leq N}f(n)\]
always exists?

OPEN

Is there an infinite set of primes $P$ such that if $\{a_1<a_2<\cdots\}$ is the set of integers divisible only by primes in $P$ then $\lim a_{i+1}-a_i=\infty$?

Originally asked to Erdős by Wintner. The limit is infinite for a finite set of primes, which follows from a theorem of Pólya.

OPEN - $100

Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from the trivial coincidences). Is it true that
\[ f(N)\sim N^{1/3}?\]

Originally asked to Erdős by Bose. Bose and Chowla [BoCh62] provided a construction proving one half of this, namely
\[(1+o(1))N^{1/3}\leq f(N).\]
The best upper bound known to date is due to Green [Gr01],
\[f(N) \leq ((7/2)^{1/3}+o(1))N^{1/3}\]
(note that $(7/2)^{1/3}\approx 1.519\cdots$).

More generally, Bose and Chowla conjectured that the maximum size of $A\subseteq \{1,\ldots,N\}$ with all $r$-fold sums distinct (aside from the trivial coincidences) then \[\lvert A\rvert \sim N^{1/r}.\] This is known only for $r=2$ (see [30]).

OPEN

For every $n\geq 2$ there exist distinct integers $1\leq x<y<z$ such that
\[\frac{4}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]

The Erdős-Straus conjecture. The existence of a representation of $4/n$ as the sum of at most four distinct unit fractions follows trivially from a greedy algorithm.

Schinzel conjectured the generalisation that, for any fixed $a$, if $n$ is sufficiently large in terms of $a$ then there exist distinct integers $1\leq x<y<z$ such that \[\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]

SOLVED

Let $A\subseteq \mathbb{N}$ be an infinite set such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that
\[\limsup_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}\geq 3?\]

Erdős writes it is 'easy to see' that this holds with $3$ replaced by $2$, and that $3$ would be best possible here. We do not see an easy argument that this holds with $2$, but this follows e.g. from the main result of Mann [Ma60].

The answer is yes, proved by Freiman [Fr73].

SOLVED

Let $a_1<a_2<\cdots $ be an infinite sequence of integers such that $a_{i+1}/a_i\to 1$. If every arithmetic progression contains infinitely many integers which are the sum of distinct $a_i$ then every sufficiently large integer is the sum of distinct $a_i$.

This was disproved by Cassels [Ca60].

OPEN

Let $A\subseteq \mathbb{N}$ be such that
\[\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty\]
and
\[\sum_{n\in A} \{ \theta n\}=\infty\]
for every $\theta\in (0,1)$, where $\{x\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.

Cassels [Ca60] proved this under the alternative hypotheses
\[\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert\gg \log\log x\]
and
\[\sum_{n\in A} \{ \theta n\}^2=\infty\]
for every $\theta\in (0,1)$.

SOLVED

Let $z_1,z_2,\ldots \in [0,1]$ be an infinite sequence, and define the discrepancy
\[D_N(I) = \#\{ n\leq N : z_n\in I\} - N\lvert I\rvert.\]
Must there exist some interval $I\subseteq [0,1]$ such that
\[\limsup_{N\to \infty}\lvert D_N(I)\rvert =\infty?\]

The answer is yes, as proved by Schmidt [Sc68], who later showed [Sc72] that in fact this is true for all but countably many intervals of the shape $[0,x]$.

Essentially the best possible result was proved by Tijdeman and Wagner [TiWa80], who proved that, for almost all intervals of the shape $[0,x)$, we have \[\limsup_{N\to \infty}\frac{\lvert D_N([0,x))\rvert}{\log N}\gg 1.\]

OPEN

Let $n\geq 1$ and $f(n)$ be maximal such that, for every $a_1\leq \cdots \leq a_n\in \mathbb{N}$ we have
\[\max_{\lvert z\rvert=1}\left\lvert \prod_{i}(1-z^{a_i})\right\rvert\geq f(n).\]
Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that
\[f(n) \geq \exp(n^{c})?\]

Erdős and Szekeres [ErSz59] proved that $\lim f(n)^{1/n}=1$ and $f(n)>\sqrt{2n}$. Erdős proved an upper bound of $f(n) < \exp(n^{1-c})$ for some constant $c>0$ with probabilistic methods. Atkinson [At61] showed that $f(n) <\exp(cn^{1/2}\log n)$ for some constant $c>0$.

This was improved to \[f(n) \leq \exp( cn^{1/3}(\log n)^{4/3})\] by Odlyzko [Od82].

If we denote by $f^*(n)$ the analogous quantity with the assumption that $a_1<\cdots<a_n$ then Bourgain and Chang [BoCh18] prove that \[f^*(n)< \exp(c(n\log n)^{1/2}\log\log n).\]

OPEN

Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?

A problem of Ostmann, sometimes known as the 'inverse Goldbach problem'. The answer is surely no. The best result in this direction is due to Elsholtz and Harper [ElHa15], who showed that if $A,B$ are such sets then for all large $x$ we must have
\[\frac{x^{1/2}}{\log x\log\log x} \ll \lvert A \cap [1,x]\rvert \ll x^{1/2}\log\log x\]
and similarly for $B$.

Elsholtz [El01] has proved there are no infinite sets $A,B,C$ such that $A+B+C$ agrees with the set of prime numbers up to finitely many exceptions.

See also [432].

SOLVED

How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean there are no solutions to $A\cup B=C$ with distinct $A,B,C\in \mathcal{F}$. Must $\lvert \mathcal{F}\rvert =o(2^n)$? Perhaps even
\[\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}?\]

The estimate $\lvert \mathcal{F}\rvert=o(2^n)$ implies that if $A\subset \mathbb{N}$ is a set of positive density then there are infinitely many distinct $a,b,c\in A$ such that $[a,b]=c$ (i.e. [487]).

Solved by Kleitman [Kl71], who proved \[\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}.\]

OPEN

For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). If $S_n(t)$ is the number of roots of $\sum_{1\leq k\leq n}\epsilon_k(t)z^k$ in $\lvert z\rvert \leq1$ then is it true that, for almost all $t\in (0,1)$,
\[\lim_{n\to \infty}\frac{S_n(t)}{n}=\frac{1}{2}?\]

SOLVED

Prove that there exists an absolute constant $c>0$ such that, whenever $\{1,\ldots,N\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\{1,\ldots,N\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\in \{1,\ldots,N\}$ are in the same colour class and $a\neq b$).

A conjecture of Roth.

Solved by Erdős, Sárközy, and Sós [ESS89], who in fact prove that there are at least \[\frac{N}{2}-O(N^{1-1/2^{k+1}})\] many even numbers which are of this form. They also prove that if $k=2$ then there are at least \[\frac{N}{2}-O(\log N)\] many even numbers which are of this form, and that $O(\log N)$ is best possible, since there is a $2$-colouring such that no power of $2$ is representable as a monochromatic sum.

A refinement of this problem appears as Problem 25 on the open problems list of Ben Green.

SOLVED

Let $f(k)$ be the minimum number of terms in $P(x)^2$, where $P\in \mathbb{Q}[x]$ ranges over all polynomials with exactly $k$ non-zero terms. Is it true that $f(k)\to\infty$ as $k\to \infty$?

First investigated by Rényi and Rédei [Re47]. Erdős [Er49b] proved that $f(k)<k^{1-c}$ for some $c>0$. The conjecture that $f(k)\to \infty$ is due to Erdős and Rényi.

This was solved by Schinzel [Sc87], who proved that \[f(k) > \frac{\log\log k}{\log 2}.\] In fact Schinzel proves lower bounds for the corresponding problem with $P(x)^n$ for any integer $n\geq 1$, where the coefficients of the polynomial can be from any field with zero or sufficiently large positive characteristic.

Schinzel and Zannier [ScZa09] have improved this to \[f(k) \gg \log k.\]

OPEN

Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let
\[B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\}.\]
Must $B$ have a logarithmic density, i.e. is it true that
\[\lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ m<x}}\frac{1}{m}\]
exists?

SOLVED

Let $A\subseteq \mathbb{N}$ have positive density. Must there exist distinct $a,b,c\in A$ such that $[a,b]=c$ (where $[a,b]$ is the lowest common multiple of $a$ and $b$)?

OPEN

Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let
\[B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}.\]
If $B=\{b_1<b_2<\cdots\}$ then is it true that
\[\lim \frac{1}{x}\sum_{b_i<x}(b_{i+1}-b_i)^2\]
exists (and is finite)?

For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős.

See also [208].

SOLVED

Let $A,B\subseteq \{1,\ldots,N\}$ be such that all the products $ab$ with $a\in A$ and $b\in B$ are distinct. Is it true that
\[\lvert A\rvert \lvert B\rvert \ll \frac{N^2}{\log N}?\]

This would be best possible, for example letting $A=[1,N/2]\cap \mathbb{N}$ and $B=\{ N/2<p\leq N: p\textrm{ prime}\}$.

See also [425].

This is true, and was proved by Szemerédi [Sz76].

SOLVED

Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). If there is a constant $c$ such that $\lvert f(n+1)-f(n)\rvert <c$ for all $n$ then must there exist some $c'$ such that
\[f(n)=c'\log n+O(1)?\]

SOLVED

Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ be infinite such that $a_{i+1}/a_i\to 1$. For any $x\geq a_1$ let
\[f(x) = \frac{x-a_i}{a_{i+1}-a_i}\in [0,1),\]
where $x\in [a_i,a_{i+1})$. Is it true that, for almost all $\alpha$, the sequence $f(\alpha n)$ is uniformly distributed in $[0,1)$?

SOLVED

Does there exist a $k$ such that every sufficiently large integer can be written in the form
\[\prod_{i=1}^k a_i - \sum_{i=1}^k a_i\]
for some integers $a_i\geq 2$?

Erdős attributes this question to Schinzel. Eli Seamans has observed that the answer is yes (with $k=2$) for a very simple reason:
\[n = 2(n+2)-(2+(n+2)).\]
There may well have been some additional constraint in the problem as Schinzel posed it, but [Er61] does not record what this is.

OPEN

Let $A\subset \mathbb{C}$ be a finite set of fixed size, for any $k\geq 1$ let
\[A_k = \{ z_1+\cdots+z_k : z_i\in A\textrm{ distinct}\}.\]
For $k>2$ does the set $A_k$ (together with the size of $A$) uniquely determine the set $A$?

A problem of Selfridge and Straus [SeSt58], who prove that this is true if $k=2$ and $\lvert A\rvert \neq 2^l$ (for $l\geq 0$). On the other hand, there are examples with two distinct $A,B$ both of size $2^l$ such that $A_2=B_2$.

More generally, they prove that $A$ is uniquely determined by $A_k$ if $n$ is divisible by a prime greater than $k$. Selfridge and Straus sound more cautious than Erdős, and it may well be that for all $k>2$ there exist $A,B$ of the same size with identical $A_k=B_k$.

(In [Er61] Erdős states this problem incorrectly, replacing sums with products. This product formulation is easily seen to be false, as observed by Steinerberger: consider the case $k=3$ and subsets of the 6th roots of unity corresponding to $\{0,1,2,4\}$ and $\{0,2,3,4\}$ (as subsets of $\mathbb{Z}/6\mathbb{Z}$). The correct problem statement can be found in the paper of Selfridge and Straus that Erdős cites.)

OPEN

Let $\alpha,\beta \in \mathbb{R}$. Is it true that
\[\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0\]
where $\|x\|$ is the distance from $x$ to the nearest integer?

The infamous Littlewood conjecture.

SOLVED

Let $\alpha \in \mathbb{R}$ be irrational and $\epsilon>0$. Are there positive integers $x,y,z$ such that
\[\lvert x^2+y^2-z^2\alpha\rvert <\epsilon?\]

Originally a conjecture due to Oppenheim. Davenport and Heilbronn [DaHe46] solve the analogous problem for quadratic forms in 5 variables.

This is true, and was proved by Margulis [Ma89].

SOLVED

How many antichains in $[n]$ are there? That is, how many families of subsets of $[n]$ are there such that, if $\mathcal{F}$ is such a family and $A,B\in \mathcal{F}$, then $A\not\subseteq B$?

Sperner's theorem states that $\lvert \mathcal{F}\rvert \leq \binom{n}{\lfloor n/2\rfloor}$. This is also known as Dedekind's problem.

Resolved by Kleitman [Kl69], who proved that the number of such families is \[2^{(1+o(1))\binom{n}{\lfloor n/2\rfloor}}.\]

SOLVED

Let $z_1,\ldots,z_n\in\mathbb{C}$ with $1\leq \lvert z_i\rvert$ for $1\leq i\leq n$. Let $D$ be an arbitrary disc of radius $1$. Is it true that the number of sums of the shape
\[\sum_{i=1}^n\epsilon_iz_i \textrm{ for }\epsilon_i\in \{-1,1\}\]
which lie in $D$ is at most $\binom{n}{\lfloor n/2\rfloor}$?

A strong form of the Littlewood-Offord problem. Erdős [Er45] proved this is true if $z_i\in\mathbb{R}$, and for general $z_i\in\mathbb{C}$ proved a weaker upper bound of
\[\ll \frac{2^n}{\sqrt{n}}.\]
This was solved in the affirmative by Kleitman [Kl65], who also later generalised this to arbitrary Hilbert spaces [Kl70].

See also [395].

SOLVED

Let $M=(a_{ij})$ be a real $n\times n$ doubly stochastic matrix (i.e. the entries are non-negative and each column and row sums to $1$). Does there exist some $\sigma\in S_n$ such that
\[\prod_{1\leq i\leq n}a_{i\sigma(i)}\geq n^{-n}?\]

A weaker form of the conjecture of van der Waerden, which states that
\[\mathrm{perm}(M)=\sum_{\sigma\in S_n}\prod_{1\leq i\leq n}a_{i\sigma(i)}\geq n^{-n}n!\]
with equality if and only if $a_{ij}=1/n$ for all $i,j$.

This conjecture is true, and was proved by Marcus and Minc [MaMi62].

Erdős also conjectured the even weaker fact that there exists some $\sigma\in S_n$ such that $a_{i\sigma(i)}\neq 0$ for all $i$ and \[\sum_{i}a_{i\sigma(i)}\geq 1.\] This weaker statement was proved by Marcus and Ree [MaRe59].

van der Waerden's conjecture itself was proved by Gyires [Gy80], Egorychev [Eg81], and Falikman [Fa81].

OPEN - $500

What is $\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of $3$-edges which can placed on $n$ vertices so that there exists no $K_4^3$, a set of 4 vertices which is covered by all 4 possible $3$-edges.

A problem of Turán. Turán observed that dividing the vertices into three equal parts $X_1,X_2,X_3$, and taking the edges to be those triples that either have exactly one vertex in each part or two vertices in $X_i$ and one vertex in $X_{i+1}$ (where $X_4=X_1$) shows that
\[\mathrm{ex}_3(n,K_4^3)\geq\left(\frac{5}{9}+o(1)\right)\binom{n}{3}.\]
This is probably the truth. The current best upper bound is
\[\mathrm{ex}_3(n,K_4^3)\leq 0.5611666\binom{n}{3},\]
due to Razborov [Ra10].

See also [712] for the general case.

OPEN

For every $x\in\mathbb{R}$ let $A_x\subset \mathbb{R}$ be a bounded set with outer measure $<1$. Must there exist an infinite independent set, that is, some infinite $X\subseteq \mathbb{R}$ such that $x\not\in A_y$ for all $x\neq y\in X$?

If the sets $A_x$ are closed and have measure $<1$, then must there exist an independent set of size $3$?

Erdős and Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets (under the assumptions in the first problem).

Erdős writes in [Er61] that Gladysz has proved the existence of an independent set of size $2$ in the second question, but I cannot find a reference.

Hechler [He72] has shown the answer to the first question is no, assuming the continuum hypothesis.

SOLVED

What is the size of the largest $A\subseteq \mathbb{R}^n$ such that there are only two distinct distances between elements of $A$? That is,
\[\# \{ \lvert x-y\rvert : x\neq y\in A\} = 2.\]

Asked to Erdős by Coxeter. Erdős thought he could show that $\lvert A\rvert \leq n^{O(1)}$, but later discovered a mistake in his proof, and his proof only gave $\leq \exp(n^{1-o(1)})$.

Bannai, Bannai, and Stanton [BBS83] have proved that \[\lvert A\rvert \leq \binom{n+2}{2}.\] A simple proof of this upper bound was given by Petrov and Pohoata [PePo21].

Shengtong Zhang has observed that a simple lower bound of $\binom{n}{2}$ is given by considering all points with exactly two coordinates equal to $1$ and all others equal to $0$.

OPEN

What is the size of the largest $A\subseteq \mathbb{R}^n$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x,y,z$ from $A$, at least two of the distances $\lvert x-y\rvert,\lvert y-z\rvert,\lvert x-z\rvert$ are equal.

When $n=2$ the answer is $6$ (due to Kelly [ErKe47]). When $n=3$ the answer is $8$ (due to Croft [Cr62]). The best upper bound known in general is due to Blokhuis [Bl84] who showed that
\[\lvert A\rvert \leq \binom{n+2}{2}.\]

Alweiss has observed a lower bound of $\binom{n+1}{2}$ follows from considering the subset of $\mathbb{R}^{n+1}$ formed of all vectors $e_i+e_j$ where $e_i,e_j$ are distinct coordinate vectors. This set can be viewed as a subset of some $\mathbb{R}^n$, and is easily checked to have the required property.

The fact that the truth for $n=3$ is $8$ suggests that neither of these bounds is the truth.

SOLVED

Let $\alpha_n$ be the infimum of all $0\leq \alpha\leq \pi$ such that in every set $A\subset \mathbb{R}^2$ of size $n$ there exist three distinct points $x,y,z\in A$ such that the angle determined by $xyz$ is at least $\alpha$. Determine $\alpha_n$.

Blumenthal's problem. Szekeres [Sz41] showed that
\[\alpha_{2^n+1}> \pi \left(1-\frac{1}{n}+\frac{1}{n(2^n+1)^2}\right)\]
and
\[\alpha_{2^n}\leq \pi\left(1-\frac{1}{n}\right).\]
Erdős and Szekeres [ErSz60] showed that
\[\alpha_{2^n}=\alpha_{2^n-1}= \pi\left(1-\frac{1}{n}\right),\]
and suggested that perhaps $\alpha_{N}=\pi(1-1/n)$ for $2^{n-1}<N\leq 2^n$. This was disproved by Sendov [Se92].

Sendov [Se93] provided the definitive answer, proving that $\alpha_N=\pi(1-1/n)$ for $2^{n-1}+2^{n-3}<N\leq 2^n$ and $\alpha_N=\pi(1-\frac{1}{2n-1})$ for $2^{n-1}<N\leq 2^{n-1}+2^{n-3}$.

SOLVED

Is every set of diameter $1$ in $\mathbb{R}^n$ the union of at most $n+1$ sets of diameter $<1$?

Borsuk's problem. This is trivially true for $n=1$ and easy for $n=2$. For $n=3$ it is true, which was proved by Eggleston [Eg55].

The answer is in fact no in general, as shown by Kahn and Kalai [KaKa93], who proved that it is false for $n>2014$. The current smallest $n$ where Borsuk's conjecture is known to be false is $n=64$, a result of Brouwer and Jenrich [BrJe14].

If $\alpha(n)$ is the smallest number of pieces of diameter $<1$ required (so Borsuk's original conjecture was that $\alpha(n)=n+1$) then Kahn and Kalai's construction shows that $\alpha(n)\geq (1.2)^{\sqrt{n}}$. The best upper bound, due to Schramm [Sc88], is that \[\alpha(n) \leq ((3/2)^{1/2}+o(1))^{n}.\]

OPEN

What is the minimum number of circles determined by any $n$ points in $\mathbb{R}^2$, not all on a circle?

There is clearly some non-degeneracy condition intended here - probably either that not all the points are on a line, or the stronger condition that no three points are on a line.

OPEN

Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Estimate $\alpha(n)$.

Heilbronn's triangle problem. It is trivial that $\alpha(n) \ll 1/n$. Erdős observed that $\alpha(n)\gg 1/n^2$. The current best bounds are
\[\frac{\log n}{n^2}\ll \alpha(n) \ll \frac{1}{n^{8/7+1/2000}}.\]
The lower bound is due to Komlós, Pintz, and Szemerédi [KPS82]. The upper bound is due to Cohen, Pohoata, and Zakharov [CPZ23] (improving on an exponent of $8/7$ due to Komlós, Pintz, and Szemerédi [KPS81]).

OPEN

What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\mathbb{R}^2$ such that no two points of the same colour are distance $1$ apart?

The Hadwiger-Nelson problem. Let $\chi$ be the chromatic number of the plane. An equilateral triangle trivially shows that $\chi\geq 3$. There are several small graphs that show $\chi\geq 4$ (in particular the Moser spindle and Golomb graph). The best bounds currently known are
\[5 \leq \chi \leq 7.\]
The lower bound is due to de Grey [dG18]. The upper bound can be seen by colouring the plane by tesselating by hexagons with diameter slightly less than $1$.

OPEN

Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set
\[\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}\]
be covered by a set of circles the sum of whose radii is $\leq 2$?

OPEN

If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that
\[\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?\]

OPEN

Let $f(z)\in \mathbb{C}[z]$ be a monic polynomial of degree $n$ and
\[A = \{ z\in \mathbb{C} : \lvert f(z)\rvert\leq 1\}.\]
Is it true that, for every such $f$ and constant $c>0$, the set $A$ can have at most $O_c(1)$ many components of diameter $>1+c$ (where the implied constant is in particular independent of $n$)?

SOLVED

Is it true that, if $A\subset \mathbb{Z}$ is a finite set of size $N$, then
\[\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,\]
where $e(x)=e^{2\pi ix }$?

OPEN

Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function. What is the greatest possible value of
\[\liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}?\]

OPEN

Let $f(z)$ be an entire function. Does there exist a path $L$ so that, for every $n$,
\[\lvert f(z)/z^n\rvert \to \infty\]
as $z\to \infty$ along $L$?

Can the length of this path be estimated in terms of $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$? Does there exist a path along which $\lvert f(z)\rvert$ tends to $\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\epsilon$)?

Boas (unpublished) has proved the first part, that such a path must exist.

OPEN

Let $f(z)$ be an entire function, not a polynomial. Does there exist a path $C$ such that, for every $\lambda>0$, the integral
\[\int_C \lvert f(z)\rvert^{-\lambda} \mathrm{d}z\]
is finite?

Huber [Hu57] proved that for every $\lambda>0$ there is a path $C_\lambda$ such that this integral is finite.

OPEN

Let $f(z)=\sum_{k\geq 1}a_k z^{n_k}$ be an entire function of finite order such that $\lim n_k/k=\infty$. Let $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$ and $m(r)=\max_n \lvert a_nr^n\rvert$. Is it true that
\[\limsup\frac{\log m(r)}{\log M(r)}=1?\]

OPEN

Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function. Is it true that if $n_k/k\to \infty$ then $f(z)$ assumes every value infinitely often?

SOLVED

Let $z_1,\ldots,z_n\in \mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that
\[\max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c?\]

A problem of Turán, who proved that this maximum is $\gg 1/n$. This was solved by Atkinson [At61b], who showed that $c=1/6$ suffices. This has been improved by Biró, first to $c=1/2$ [Bi94], and later to an absolute constant $c>1/2$ [Bi00]. Based on computational evidence it is likely that the optimal value of $c$ is $\approx 0.7$.

SOLVED

Let $f$ be a Rademacher multiplicative function: a random $\{-1,0,1\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\cdots p_r)=f(p_1)\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does there exist some constant $c>0$ such that, almost surely,
\[\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N\log\log N}}=c?\]

Note that if we drop the multiplicative assumption, and simply assign $f(m)=\pm 1$ at random, then this statement is true (with $c=\sqrt{2}$), the law of the iterated logarithm.

Wintner [Wi44] proved that, almost surely, \[\sum_{m\leq N}f(m)\ll N^{1/2+o(1)},\] and Erdős improved the right-hand side to $N^{1/2}(\log N)^{O(1)}$. Lau, Tenenbaum, and Wu [LTW13] have shown that, almost surely, \[\sum_{m\leq N}f(m)\ll N^{1/2}(\log\log N)^{2+o(1)}.\] Harper [Ha13] has shown that the sum is almost surely not $O(N^{1/2}/(\log\log N)^{5/2+o(1)})$, and conjectured that in fact Erdős' conjecture is false, and almost surely \[\sum_{m\leq N}f(m) \ll N^{1/2}(\log\log N)^{1/4+o(1)}.\] This was proved by Caich [Ca23].

OPEN

For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). Let $R_n(t)$ denote the number of real roots of $\sum_{1\leq k\leq n}\epsilon_k(t)z^k$. Is it true that, for almost all $t\in (0,1)$, we have
\[\lim_{n\to \infty}\frac{R_n(t)}{\log n}=\frac{\pi}{2}?\]

SOLVED

Is it true that all except $o(2^n)$ many polynomials of degree $n$ with $\pm 1$-valued coefficients have $(\frac{1}{2}+o(1))n$ many roots in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$?

Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.

Solved by Yakir [Ya21], who proved that almost all such polynomials have \[\frac{n}{2}+O(n^{9/10})\] many roots in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$.

OPEN

For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). Does there exist some constant $C>0$ such that, for almost all $t\in (0,1)$,
\[\max_{\lvert z\rvert=1}\left\lvert \sum_{k\leq n}\epsilon_k(t)z^k\right\rvert=(C+o(1))\sqrt{n\log n}?\]

Salem and Zygmund [SZ54] proved that $\sqrt{n\log n}$ is the right order of magnitude, but not an asymptotic.

OPEN

For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for almost all $t\in(0,1)$) for
\[M_n(t)=\max_{x\in [0,1]}\left\lvert \sum_{k\leq n}\epsilon_k(t)x^k\right\rvert?\]

A problem of Salem and Zygmund [SZ54]. Chung showed that, for almost all $t$, there exist infinitely many $n$ such that
\[M_n(t) \ll \left(\frac{n}{\log\log n}\right)^{1/2}.\]
Erdős (unpublished) showed that for almost all $t$ and every $\epsilon>0$ we have $\lim_{n\to \infty}M_n(t)/n^{1/2-\epsilon}=\infty$.

SOLVED

Is it true that all except at most $o(2^n)$ many degree $n$ polynomials with $\pm 1$-valued coefficients $f(z)$ have $\lvert f(z)\rvert <1$ for some $\lvert z\rvert=1$?
What is the behaviour of
\[m(f)=\min_{\lvert z\rvert=1}\lvert f(z)\rvert?\]

Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Radamacher coefficients, i.e. independent uniform $\pm 1$ values. The first problem asks whether $m(f)<1$ almost surely. Littlewood [Li66] conjectured that the stronger $m(f)=o(1)$ holds almost surely.

The answer to both questions is yes: Littlewood's conjecture was solved by Kashin [Ka87], and Konyagin [Ko94] improved this to show that $m(f)\leq n^{-1/2+o(1)}$ almost surely. This is essentially best possible, since Konyagin and Schlag [KoSc99] proved that for any $\epsilon>0$ \[\limsup_{n\to \infty} \mathbb{P}(m(f) \leq \epsilon n^{-1/2})\ll \epsilon.\] Cook and Nguyen [CoNg21] have identified the limiting distribution, proving that for any $\epsilon>0$ \[\lim_{n\to \infty} \mathbb{P}(m(f) > \epsilon n^{-1/2}) = e^{-\epsilon \lambda}\] where $\lambda$ is an explicit constant.

SOLVED

Let $a_n\geq 0$ with $a_n\to 0$ and $\sum a_n=\infty$. Find a necessary and sufficient condition on the $a_n$ such that, if we choose (independently and uniformly) random arcs on the unit circle of length $a_n$, then all the circle is covered with probability $1$.

A problem of Dvoretzky [Dv56]. It is easy to see that (under the given conditions alone) almost all the circle is covered with probability $1$.

Kahane [Ka59] showed that $a_n=\frac{1+c}{n}$ with $c>0$ has this property, which Erd\H{s} (unpublished) improved to $a_n=\frac{1}{n}$. Erd\{o}s also showed that $a_n=\frac{1-c}{n}$ with $c>0$ does not have this property.

Solved by Shepp [Sh72], who showed that a necessary and sufficient condition is that \[\sum_n \frac{e^{a_1+\cdots+a_n}}{n^2}=\infty.\]

OPEN

Let $a_n\in \mathbb{R}$ be such that $\sum_n \lvert a_n\rvert^2=\infty$ and $\lvert a_n\rvert=o(1/\sqrt{n})$. Is it true that, for almost all $\epsilon_n=\pm 1$, there exists some $z$ with $\lvert z\rvert=1$ (depending on the choice of signs) such that
\[\sum_n \epsilon_n a_n z^n\]
converges?

It is unclear to me whether Erdős also intended to assume that $\lvert a_{n+1}\rvert\leq \lvert a_n\rvert$.

It is 'well known' that, for almost all $\epsilon_n=\pm 1$, the series diverges for almost all $\lvert z\rvert=1$ (assuming only $\sum \lvert a_n\rvert^2=\infty$).

Dvoretzky and Erdős [DE59] showed that if $\lvert a_n\rvert >c/\sqrt{n}$ then, for almost all $\epsilon_n=\pm 1$, the series diverges for all $\lvert z\rvert=1$.

OPEN

Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine
\[C_k=\lim_{n\to\infty}f(n,k)^{1/n}.\]

The constant $C_k$ is sometimes known as the connective constant. Hammersley and Morton [HM54] showed that this limit exists, and it is trivial that $k\leq C_k\leq 2k-1$.

Kesten [Ke63] proved that $C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade [CLS07].

Conway and Guttmann [CG93] showed that $C_2\geq 2.62$ and Alm [Al93] showed that $C_2\leq 2.696$. Jacobsen, Scullard, and Guttmann [JSG16] have computed the first few decimal places of $C_2$, showing that \[C_2 = 2.6381585303279\cdots.\]

OPEN

Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\mathbb{Z}^k$ (conditional on no self intersections). Is it true that
\[\lim_{n\to \infty}\frac{d_2(n)}{n^{1/2}}= \infty?\]
Is it true that
\[d_k(n)\ll n^{1/2}\]
for $k\geq 3$?

OPEN - $500

Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that
\[\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?\]
Or even $\gg n/\sqrt{\log n}$?

The pinned distance problem, a stronger form of [89]. The example of an integer grid show that $n/\sqrt{\log n}$ would be best possible.

It may be true that there are $\gg n$ many such points, or that this is true on average. In [Er97e] Erdős offers \$500 for a solution to this problem, but it is unclear whether he intended this for proving the existence of a single such point or for $\gg n$ many such points.

In [Er97e] Erdős wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see [89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of $n^{o(1)}$.

The best known bound is \[\gg n^{c-o(1)},\] due to Katz and Tardos [KaTa04], where \[c=\frac{48-14e}{55-16e}=0.864137\cdots.\]

OPEN

If $\pi(x)$ counts the number of primes in $[1,x]$ then is it true that
\[\pi(x+y) \leq \pi(x)+\pi(y)?\]

Commonly known as the second Hardy-Littlewood conjecture. In [Er85c] Erdős describes it as 'an old conjecture of mine which was probably already stated by Hardy and Littlewood'.

This is probably false, since Hensley and Richards [HeRi73] have shown that this is false assuming the Hardy-Littlewood prime tuples conjecture.

Erdős [Er85c] reports Straus as remarking that the 'correct way' of stating this conjecture would have been \[\pi(x+y) \leq \pi(x)+2\pi(y/2).\] Clark and Jarvis [ClJa01] have shown this is also incompatible with the prime tuples conjecture.

In [Er85c] Erdős conjectures the weaker result (which in particular follows from the conjecture of Straus) that \[\pi(x+y) \leq \pi(x)+\pi(y)+O\left(\frac{y}{(\log y)^2}\right),\] which the Hensley and Richards result shows (conditionally) would be best possible. Richards conjectured that this is false.

Erdős and Richards further conjectured that the original inequality is true almost always - that is, the set of $x$ such that $\pi(x+y)\leq \pi(x)+\pi(y)$ for all $y<x$ has density $1$. They could only prove that this set has positive lower density.

They also conjectured that for every $x$ the inequality $\pi(x+y)\leq \pi(x)+\pi(y)$ is true provided $y \gg (\log x)^C$ for some large constant $C>0$.

Hardy and Littlewood proved \[\pi(x+y) \leq \pi(x)+O(\pi(y)).\] The best known in this direction is a result of Montgomery and Vaughan [MoVa73], which shows \[\pi(x+y) \leq \pi(x)+2\frac{y}{\log y}.\]