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of the constant have been given (see [St23] for a history), with the current record $\sqrt{2/\pi}$ first proved in unpublished work of Elkies and Gleason.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].

OPEN - $5000

If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions?

This is essentially asking for good bounds on $r_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ without a non-trivial $k$-term arithmetic progression. For example, a bound like
\[r_k(N) \ll_k \frac{N}{(\log N)(\log\log N)^2}\]
would be sufficient.

Even the case $k=3$ is non-trivial, but was proved by Bloom and Sisask [BlSi20]. Much better bounds for $r_3(N)$ were subsequently proved by Kelley and Meka [KeMe23]. Green and Tao [GrTa17] proved $r_4(N)\ll N/(\log N)^{c}$ for some small constant $c>0$. Gowers [Go01] proved \[r_k(N) \ll \frac{N}{(\log\log N)^{c_k}},\] where $c_k>0$ is a small constant depending on $k$. The current best bounds for general $k$ are due to Leng, Sah, and Sawhney [LSS24], who show that \[r_k(N) \ll \frac{N}{\exp((\log\log N)^{c_k})}\] for some constant $c_k>0$ depending on $k$.

Curiously, Erdős [Er83c] thought this conjecture was the 'only way to approach' the conjecture that there are arbitrarily long arithmetic progressions of prime numbers, now a theorem due to Green and Tao [GrTa08] (see [219]).

In [Er81] Erdős makes the stronger conjecture that \[r_k(N) \ll_C\frac{N}{(\log N)^C}\] for every $C>0$ (now known for $k=3$ due to Kelley and Meka [KeMe23]) - see [140].

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

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 $n$.

OPEN - $500

Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial coincidences). Is it true that
\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?\]

Erdős proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then
\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0.\]

OPEN - $100

If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that
\[ \binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1),\]
where $f(N)$ is the maximum possible size of a Sidon set in $\{1,\ldots,N\}$? If $\lvert A\rvert=\lvert B\rvert$ then can this bound be improved to
\[\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq (1-c)\binom{f(N)}{2}\]
for some constant $c>0$?

OPEN

Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M=M(\epsilon)$ and $B\subset \{N+1,\ldots,M\}$ such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$?

See also [707].

OPEN - $250

Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$
\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?\]

The sum-product problem. Erdős and Szemerédi [ErSz83] proved a lower bound of $\lvert A\rvert^{1+c}$ for some constant $c>0$, and an upper bound of
\[\lvert A\rvert^2 \exp\left(-c\frac{\log\lvert A\rvert}{\log\log \lvert A\rvert}\right)\]
for some constant $c>0$. The lower bound has been improved a number of times. The current record is
\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{1558}{1167}-o(1)}\]
due to Rudnev and Stevens [RuSt22] (note $1558/1167=1.33504\cdots$).

There is likely nothing special about the integers in this question, and indeed Erdős and Szemerédi also ask a similar question about finite sets of real or complex numbers. The current best bound for sets of reals is the same bound of Rudnev and Stevens above. The best bound for complex numbers is \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{5}{4}},\] due to Solymosi [So05].

One can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that if $A\subseteq \mathbb{F}_p$ with $\lvert A\rvert <p^{5/8}$ then \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{11}{9}+o(1)},\] due to Rudnev, Shakan, and Shkredov [RSS20].

There is also a natural generalisation to higher-fold sum and product sets. For example, in [ErSz83] (and in [Er91]) Erdős and Szemerédi also conjecture that for any $m\geq 2$ and finite set of integers $A$ \[\max( \lvert mA\rvert,\lvert A^m\rvert)\gg \lvert A\rvert^{m-o(1)}.\] See [53] for more on this generalisation and [808] for a stronger form of the original conjecture. See also [818] for a special case.

SOLVED

Let $A$ be a finite set of integers. Is it true that, for every $k$, if $\lvert A\rvert$ is sufficiently large depending on $k$, then there are least $\lvert A\rvert^k$ many integers which are either the sum or product of distinct elements of $A$?

Asked by Erdős and Szemerédi [ErSz83]. Solved in this form by Chang [Ch03].

Erdős and Szemerédi proved that there exist arbitrarily large sets $A$ such that the integers which are the sum or product of distinct elements of $A$ is at most \[\exp\left(c (\log \lvert A\rvert)^2\log\log\lvert A\rvert\right)\] for some constant $c>0$.

See also [52].

SOLVED

Any $A\subseteq \mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and $C$ are infinite.

The Erdős sumset conjecture. Proved by Moreira, Richter, and Robertson [MRR19].

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].

SOLVED - $500

Let $r_3(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain a non-trivial $3$-term arithmetic progression. Prove that $r_3(N)\ll N/(\log N)^C$ for every $C>0$.

OPEN - $10000

Let $r_k(N)$ be the largest possible size of a subset of $\{1,\ldots,N\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove an asymptotic formula for $r_k(N)$.

Erdős remarked this is 'probably unattackable at present'. In [Er97c] Erdős offered \$1000, but given that he elsewhere offered \$5000 just for (essentially) showing that $r_k(N)=o_k(N/\log N)$, that value seems odd. In [Er81] he offers \$10000, stating it is 'probably enormously difficult'.

The best known upper bounds for $r_k(N)$ are due to Kelley and Meka [KeMe23] for $k=3$, Green and Tao [GrTa17] for $k=4$, and Leng, Sah, and Sawhney [LSS24] for $k\geq 5$. An asymptotic formula is still far out of reach, even for $k=3$.

OPEN

Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.

Investigated by Erdős and Freud. This has been discussed on MathOverflow, where LeechLattice shows
\[h(N) \ll N^{2/3}.\]
The observation of Zachary Hunter in that question coupled with the bounds of Kelley-Meka [KeMe23] imply that
\[h(N) \gg \exp(c(\log N)^{1/12})\]
for some $c>0$.

OPEN

Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is
\[ \lim_{N\to \infty}\frac{F(N)}{N}?\]
Is this limit irrational?

This limit was proved to exist by Graham, Spencer, and Witsenhausen [GrSpWi77]. Similar questions can be asked for the density or upper density of infinite sets without such configurations.

OPEN

Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Estimate $f(k)$.

Is \[\lim_{k\to \infty}\frac{f(k)}{\log W(k)}=\infty\] where $W(k)$ is the van der Waerden number?

Gerver [Ge77] has proved
\[f(k) \geq (1+o(1))k\log k.\]
It is trivial that
\[\frac{f(k)}{\log W(k)}\geq \frac{1}{2},\]
but improving the right-hand side to any constant $>1/2$ is open.

OPEN

Let $F(N)$ be the smallest possible size of $A\subset \{0,1,\ldots,N\}$ such that $\{0,1,\ldots,N\}\subset A-A$. Find the value of
\[\lim_{N\to \infty}\frac{F(N)}{N^{1/2}}.\]

The Sparse Ruler problem. Rédei asked whether this limit exists, which was proved by Erdős and Gál [ErGa48]. Bounds on the limit were improved by Leech [Le56]. The limit is known to be in the interval $[1.56,\sqrt{3}]$. The lower bound is due to Leech [Le56], the upper bound is due to Wichmann [Wi63]. Computational evidence by Pegg [Pe20] suggests that the upper bound is the truth. A similar question can be asked without the restriction $A\subset \{0,1,\ldots,N\}$.

SOLVED

Is it true that for every $\epsilon>0$ and integer $t\geq 1$, if $N$ is sufficiently large and $A$ is a subset of $[t]^N$ of size at least $\epsilon t^N$ then $A$ must contain a combinatorial line $P$ (a set $P=\{p_1,\ldots,p_t\}$ where for each coordinate $1\leq j\leq t$ the $j$th coordinate of $p_i$ is either $i$ or constant).

OPEN

Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?

First asked by Hindman. Hindman [Hi80] has proved this is false (with 7 colours) if we ask for an infinite $A$.

Moreira [Mo17] has proved that in any finite colouring of $\mathbb{N}$ there exist $x,y$ such that $\{x,x+y,xy\}$ are all the same colour.

Alweiss [Al23] has proved that, in any finite colouring of $\mathbb{Q}\backslash \{0\}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour. Bowen and Sabok [BoSa22] had proved this earlier for the first non-trivial case of $\lvert A\rvert=2$.

OPEN

Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$ such that
\[ \left\lvert \sum_{n\in P}f(n)\right\rvert\geq \ell.\]
Find good upper bounds for $N(k,\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that
\[N(k,ck)\leq C^k?\]
What about
\[N(k,2)\leq C^k\]
or
\[N(k,\sqrt{k})\leq C^k?\]

Spencer [Sp73] has proved that if $k=2^tm$ with $m$ odd then
\[N(k,1)=2^t(k-1)+1.\]
Erdős and Graham write that 'no decent bound' is known even for $N(k,2)$. Probabilistic methods imply that, for every fixed constant $c>0$, we have $N(k,ck)>C_c^k$ for some $C_c>1$.

SOLVED

Let $1\leq k<\ell$ be integers and define $F_k(N,\ell)$ to be minimal such that every set $A\subset \mathbb{N}$ of size $N$ which contains at least $F_k(N,\ell)$ many $k$-term arithmetic progressions must contain an $\ell$-term arithmetic progression. Find good upper bounds for $F_k(N,\ell)$. Is it true that
\[F_3(N,4)=o(N^2)?\]
Is it true that for every $\ell>3$
\[\lim_{N\to \infty}\frac{\log F_3(N,\ell)}{\log N}=2?\]

Erdős remarks the upper bound $o(N^2)$ is certainly false for $\ell >\epsilon \log N$. The answer is yes: Fox and Pohoata [FoPo20] have shown that, for all fixed $1\leq k<\ell$,
\[F_k(N,\ell)=N^{2-o(1)}\]
and in fact
\[F_{k}(N,\ell) \leq \frac{N^2}{(\log\log N)^{C_\ell}}\]
where $C_\ell>0$ is some constant. In fact, they show that, if $r_\ell(N)$ is the size of the largest subset of $\{1,\ldots,N\}$ without an $\ell$-term arithmetic progression then there exists some absolute constant $c>0$ such that
\[\left(c \frac{r_\ell(N)}{N}\right)^{2(k-1)}N^2 < F_k(N,\ell) <\left(\frac{r_\ell(N)}{N}\right)^{O(1)}N^2.\]
Any improved bounds for Szemerédi's theorem (see [139]) therefore yield improved bounds for $F_k(N,\ell)$. In particular, the bounds of Leng, Sah, and Sawhney [LSS24] imply
\[F_k(N,\ell) \leq \frac{N^2}{\exp((\log\log N)^{c_\ell})}\]
for some constant $c_\ell>0$.

SOLVED

Let $f_3(n)$ be the maximal size of a subset of $\{0,1,2\}^n$ which contains no three points on a line. Is it true that $f_3(n)=o(3^n)$?

Originally considered by Moser. It is trivial that $f_3(n)\geq R_3(3^n)$, the maximal size of a subset of $\{1,\ldots,3^n\}$ without a three-term arithmetic progression. Moser showed that
\[f_3(n) \gg \frac{3^n}{\sqrt{n}}.\]

The answer is yes, which is a corollary of the density Hales-Jewett theorem, proved by Furstenberg and Katznelson [FuKa91].

OPEN

Let $F(N)$ be the maximal size of $A\subseteq \{1,\ldots,N\}$ which is 'non-averaging', so that no $n\in A$ is the arithmetic mean of at least two elements in $A$. What is the order of growth of $F(N)$?

Originally due to Straus. It is known that
\[N^{1/4}\ll F(N) \ll N^{\sqrt{2}-1+o(1)}.\]
The lower bound is due to Bosznay [Bo89] and the upper bound to Conlon, Fox, and Pham [CFP23] (improving on earlier bound due to Erdős and Sárközy [ErSa90] of $\ll (N\log N)^{1/2}$).

See also [789].

OPEN

Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$.

Originally asked by Cohen. Erdős observed that colouring according to whether $\{ \sqrt{2}n\}<1/2$ or not implies $f(d) \ll d$ (using the fact that $\|\sqrt{2}q\| \gg 1/q$ for all $q$, where $\|x\|$ is the distance to the nearest integer). Beck [Be80] has improved this using the probabilistic method, constructing a colouring that shows $f(d)\leq (1+o(1))\log_2 d$. Van der Waerden's theorem implies $f(d)\to \infty$ is necessary.

OPEN

Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\{1,\ldots,N\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progression or a rainbow arithmetic progression (i.e. all elements are different colours). Estimate $H(k)$. Is it true that
\[H(k)^{1/k}/k \to \infty\]
as $k\to\infty$?

This type of problem belongs to 'canonical' Ramsey theory. The existence of $H(k)$ follows from Szemerédi's theorem, and it is easy to show that $H(k)^{1/k}\to\infty$.

OPEN

Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ without a $k$-term arithmetic progression? Is it true that
\[\lim_{N\to \infty}\frac{R_3(N)}{G_3(N)}=1?\]

First asked and investigated by Riddell [Ri69]. It is trivial that $G_k(N)\leq R_k(N)$, and it is possible that $G_k(N) <R_k(N)$ (for example with $k=3$ and $N=14$). Komlós, Sulyok, and Szemerédi [KSS75] have shown that $R_k(N) \ll_k G_k(N)$.

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]).

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].

OPEN

For any $n$, let $A(n)=\{0<n<\cdots\}$ be the infinite sequence with $a_0=0$ and $a_1=n$, and for $k\geq 1$ we define $a_{k+1}$ as the least integer such that there is no three-term arithmetic progression in $\{a_0,\ldots,a_{k+1}\}$.

Can the $a_k$ be explicitly determined? How fast do they grow?

It is easy to see that $A(1)$ is the set of integers which have no 2 in their base 3 expansion. Odlyzko and Stanley have found similar characterisations are known for $A(3^k)$ and $A(2\cdot 3^k)$ for any $k\geq 0$, see [OdSt78], and conjectured in general that such a sequence always eventually either satisfies
\[a_k\asymp k^{\log_23}\]
or
\[a_k \asymp \frac{k^2}{\log k}.\]
There is no known sequence which satisfies the second growth rate, but Lindhurst [Li90] gives data which suggests that $A(4)$ has such growth ($A(4)$ is given as A005487 in the OEIS).

Moy [Mo11] has proved that, for all such sequences, for all $\epsilon>0$, $a_k\leq (\frac{1}{2}+\epsilon)k^2$ for all sufficiently large $k$.

In general, sequences which begin with some initial segment and thereafter are continued in a greedy fashion to avoid three-term arithmetic progressions are known as Stanley sequences.

OPEN

Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic progression for all $i\neq j$?

Simonovits and Sós [SiSo81] have shown that $t\ll N^2$. It is possible that the maximal $t$ is achieved when we take the $A_i$ to be all arithmetic progressions in $\{1,\ldots,N\}$ containing some fixed element.

If we drop the non-empty requirement then Simonovits, Sós, and Graham [SiSoGr80] have shown that \[t\leq \binom{N}{3}+\binom{N}{2}+\binom{N}{1}+1\] and this is best possible.

SOLVED

Suppose $A\subseteq\mathbb{N}$ and $C>0$ is such that $1_A\ast 1_A(n)\leq C$ for all $n\in\mathbb{N}$. Can $A$ be partitioned into $t$ many subsets $A_1,\ldots,A_t$ (where $t=t(C)$ depends only on $C$) such that $1_{A_i}\ast 1_{A_i}(n)<C$ for all $1\leq i\leq t$ and $n\in \mathbb{N}$?

Asked by Erdős and Newman. Nešetřil and Rödl have shown the answer is no for all $C$ (source is cited as 'personal communication' in [ErGr80]). Erdős had previously shown the answer is no for $C=3,4$ and infinitely many other values of $C$.

SOLVED

Let $A,B\subseteq \mathbb{N}$ such that for all large $N$
\[\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}\]
and
\[\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}.\]
Is it true that there are infinitely many solutions to $a_1-a_2=b_1-b_2\neq 0$ with $a_1,a_2\in A$ and $b_1,b_2\in B$?

Ruzsa has observed that there is a simple counterexample: take $A$ to be the set of numbers whose binary representation has only non-zero digits in even places, and $B$ similarly but with non-zero digits only in odd places. It is easy to see $A$ and $B$ both grow like $\gg N^{1/2}$ and yet for any $n\geq 1$ there is exactly one solution to $n=a+b$ with $a\in A$ and $b\in B$.

Ruzsa suggests that a non-trivial variant of this problem arises if one imposes the stronger condition that \[\lvert A\cap \{1,\ldots,N\}\rvert \sim c_AN^{1/2}\] for some constant $c_A>0$ as $N\to \infty$, and similarly for $B$.

OPEN

Let $d(A)$ denote the density of $A\subseteq \mathbb{N}$. Characterise those $A,B\subseteq \mathbb{N}$ with positive density such that
\[d(A+B)=d(A)+d(B).\]

One way this can happen is if there exists $\theta>0$ such that
\[A=\{ n>0 : \{ n\theta\} \in X_A\}\textrm{ and }B=\{ n>0 : \{n\theta\} \in X_B\}\]
where $\{x\}$ denotes the fractional part of $x$ and $X_A,X_B\subseteq \mathbb{R}/\mathbb{Z}$ are such that $\mu(X_A+X_B)=\mu(X_A)+\mu(X_B)$. Are all possible $A$ and $B$ generated in a similar way (using other groups)?

SOLVED

Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that
\[\lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?\]

The answer is no, and a counterexample was provided by Turjányi [Tu84]. This was generalised (to the replacement of $A+A$ by the $h$-fold sumset $hA$ for any $h\geq 2$) by Ruzsa and Turjányi [RT85]. Ruzsa and Turjányi do prove (under the same hypotheses) that
\[\lim_{N\to \infty}\frac{\lvert (A+A+A)\cap \{1,\ldots,3N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty,\]
and conjecture that the same should be true with $(A+A)\cap \{1,\ldots,2N\}$ in the numerator.

OPEN

Let $A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property. What is the order of growth of $A$? Is it true that
\[\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2-\epsilon}\]
for all $\epsilon>0$ and large $N$?

Erdős and Graham [ErGr80] also ask about the difference set $A-A$, whether this has positive density, and whether this contains $22$.

This sequence is at OEIS A005282.

SOLVED

If $A\subset\mathbb{N}$ is a finite set of integers all of whose subset sums are distinct then
\[\sum_{n\in A}\frac{1}{n}<2.\]

OPEN

Let $p$ be a prime. Given any finite set $A\subseteq \mathbb{F}_p\backslash \{0\}$, is there always a rearrangement $A=\{a_1,\ldots,a_t\}$ such that all partial sums $\sum_{1\leq k\leq m}a_{k}$ are distinct, for all $1\leq m\leq t$?

A problem of Graham, who proved it when $t=p-1$. A similar conjecture was made for arbitrary abelian groups by Alspach. Such an ordering is often called a valid ordering.

This has been proved for $t\leq 12$ (see Costa and Pellegrini [CoPe20] and the references therein) and for $p-3\leq t\leq p-1$ (see Hicks, Ollis, and Schmitt [HOS19] and the references therein). Kravitz [Kr24] has proved this for \[t \leq \frac{\log p}{\log\log p}.\]

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.

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.)

SOLVED

If $\mathbb{N}$ is 2-coloured then must there exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$?

Erdös writes 'perhaps this is easy or false'. It is not true for four-term arithmetic progressions: colour the integers in $[3^{2k},3^{2k+1})$ red and all others blue.

Ryan Alweiss has provided the following simple argument showing that the answer is yes: suppose we have some red/blue colouring without this property. Without loss of generality, suppose $1$ is coloured red, and then either $3$ or $5$ must be blue.

Suppose first that $3$ is blue. If $n\geq 6$ is red then (considering $1,n,2n-1$) we deduce $2n-1$ is blue, and then (considering $3,n+1,2n-1$) we deduce that $n+1$ is red. In particular the colouring must be eventually constant, and we are done.

Now suppose that $5$ is blue. Arguing similarly (considering $1,n,2n-1$ and $5,n+2,2n-1$) we deduce that if $n\geq 8$ is red then $n+2$ is also red, and we are similarly done, since the colouring must be eventually constant on some congruence class modulo $2$.

SOLVED

Let $\delta>0$ and $N$ be sufficiently large depending on $\delta$. Is it true that if $A\subseteq \{1,\ldots,N\}^2$ has $\lvert A\rvert \geq \delta N^2$ then $A$ must contain the vertices of a square?

A problem of Graham, if the square is restricted to be axis-aligned. (It is unclear whether in [Er97e] had this restriction in mind.)

This qualitative statement follows from the density Hales-Jewett theorem proved by Furstenberg and Katznelson [FuKa91]. A quantitative proof (yet with very poor bounds) was given by Solymosi [So04].

SOLVED

Let $W(3,k)$ be the van der Waerden number defined as the minimum $n$ such that in any red/blue colouring of $\{1,\ldots,n\}$ there exists either a red $3$-term arithmetic progression or a blue $k$-term arithmetic progression.

Give reasonable bounds for $W(3,k)$. In particular, give any non-trivial lower bounds for $W(3,k)$ and prove that $W(3,k) < \exp(k^c)$ for some constant $c<1$.

While we do not have a full understanding of the growth of $W(3,k)$, both of the specific challenges of Erdős have been met.

Green [Gr22] established the superpolynomial lower bound \[W(3,k) \geq \exp\left( c\frac{(\log k)^{4/3}}{(\log\log k) ^{1/3}}\right)\] for some constant $c>0$ (in particular disproving a conjecture of Graham that $W(3,k)\ll k^2$). Hunter [Hu22] improved this to \[W(3,k) \geq \exp\left( c\frac{(\log k)^{2}}{\log\log k}\right).\] The first to show that $W(3,k) < \exp(k^c)$ for some $c<1$ was Schoen [Sc21]. The best upper bound currently known is \[W(3,k) \ll \exp\left( O((\log k)^9)\right),\] which follows from the best bounds known for sets without three-term arithmetic progressions (see [BlSi23] which improves slightly on the bounds due to Kelley and Meka [KeMe23]).

OPEN

Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive density?

Is there a basis $A$ of order $2$ such that if $A=A_1\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?

A problem of Burr and Erdős. Erdős [Er94b] thought he could construct a basis as in the second question, but 'could never quite finish the proof'.

SOLVED

Let $A\subseteq \mathbb{N}$. Can there exist some constant $c>0$ such that
\[\sum_{n\leq N} 1_A\ast 1_A(n) = cN+O(1)?\]

A conjecture of Erdős and Turán. Erdős and Fuchs [ErFu56] proved that the answer is no in a strong form: in fact even
\[\sum_{n\leq N} 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)\]
is impossible. The error term here was improved to $N^{1/4}$ by Jurkat (unpublished) and Montgomery and Vaughan [MoVa90].

SOLVED

Let $A\subseteq \mathbb{N}$. Can there exist some constant $c>0$ such that
\[\sum_{n\leq N} 1_A\ast 1_A\ast 1_A(n) = cN+O(1)?\]

The case of $1_A\ast 1_A(n)$ is the subject of [763].

The answer is no, proved in a strong form by Vaughan [Va72], who showed that in fact \[\sum_{n\leq N} 1_A\ast 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)\] is impossible. Vaughan proves a more general result that applies to any $h$-fold convolution, with different main terms permitted.

SOLVED

Let $k\geq 1$ and $H_k(n)$ be the maximal $r$ such that if $A\subset\mathbb{N}$ has $\lvert A\rvert=n$ and $\| 1_A\ast 1_A\|_\infty \leq k$ then $A$ contains a Sidon set of size at least $r$.

Is it true that $H_k(n)/n^{1/2}\to \infty$? Or even $H_k(n) > n^{1/2+c}$ for some constant $c>0$?

Erdős [Er84d] proved that
\[H_k(n) \ll n^{2/3}\]
(where the implied constant is absolute). The lower bound $H_k(n)\gg n^{1/2}$ follows from the fact that any set of size $n$ contains a Sidon set of size $\gg n^{1/2}$ (see [530]).

The answer is yes, and in fact \[H_k(n) \gg_k n^{2/3},\] proved by Alon and Erdős [AlEr85]. We sketch their proof as follows: take a random subset $A'\subset A$, including each $n\in A'$ with probability $\asymp n^{-1/3}$. The number of non-trivial additive quadruples in $A$ is $\ll n^2$ and hence only $\ll n^{2/3}$ non-trivial additive quadruples remain in $A'$. Since the size of the random subset is $\gg n^{2/3}$, all of the remaining non-trivial additive quadruples can be removed by removing at most $\lvert A'\rvert/2$ (choosing the constants suitably).

SOLVED

Let $f(k)$ be the minimal $n$ such that any $2$-colouring of $\{1,\ldots,n\}$ contains a monochromatic $k$-term descending wave: a sequence $x_1<\cdots <x_k$ such that, for $1<j<k$,
\[x_j \geq \frac{x_{j+1}+x_{j-1}}{2}.\]
Estimate $f(k)$. In particular is it true that $f(k)=k^2-k+1$ for all $k$?

SOLVED

Let $A,B\subseteq \mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\lvert A\cap [1,x]\rvert$ and similarly for $B(x)$. Is it true that if $A(x)B(x)\sim x$ then
\[A(x)B(x)-x\to \infty\]
as $x\to \infty$?

A conjecture of Erdős and Danzer. Such sets $A$ and $B$ (with all large integers in $A+B$ and $A(x)B(x)\sim x$) are called exact additive complements. Danzer [Da64] proved that exact additive complements exist.

The answer is yes, proved by Sárközy and Szemerédi [SaSz94]. Ruzsa [Ru17] has constructed, for any function $w(x)\to \infty$, such a pair of sets with \[A(x)B(x)-x<w(x)\] for infinitely many $x$.

OPEN

Let $g(n)$ be maximal such that given any set $A\subset \mathbb{R}$ with $\lvert A\rvert=n$ there exists some $B\subseteq A$ of size $\lvert B\rvert\geq g(n)$ such that $b_1+b_2\not\in A$ for all $b_1\neq b_2\in B$.

Estimate $g(n)$.

A conjecture of Erdős and Moser. Klarner proved $g(n) \gg \log n$ (indeed, a greedy construction suffices). Choi [Ch71] proved $g(n) \ll n^{2/5+o(1)}$. The current best bounds known are
\[(\log n)^{1+c} \ll g(n) \ll \exp(\sqrt{\log n})\]
for some constant $c>0$, the lower bound due to Sanders [Sa21] and the upper bound due to Ruzsa [Ru05].

OPEN

Let $f(n)$ be maximal such that if $B\subset (2n,4n)\cap \mathbb{N}$ there exists some $C\subset (n,2n)\cap \mathbb{N}$ such that $c_1+c_2\not\in B$ for all $c_1\neq c_2\in C$ and $\lvert C\rvert+\lvert B\rvert \geq f(n)$.

Estimate $f(n)$. In particular is it true that $f(n)\leq n^{1/2+o(1)}$?

A conjecture of Choi [Ch71], who proved $f(n) \ll n^{3/4}$.

OPEN

Let $h(n)$ be maximal such that if $A\subseteq \mathbb{Z}$ with $\lvert A\rvert=n$ then there is $B\subseteq A$ with $\lvert B\rvert \geq h(n)$ such that if $a_1+\cdots+a_r=b_1+\cdots+b_s$ with $a_i,b_i\in B$ then $r=s$.

Estimate $h(n)$.

OPEN

Let $l(n)$ be maximal such that if $A\subset\mathbb{Z}$ with $\lvert A\rvert=n$ then there exists some $B\subseteq A$ with $\lvert B\rvert \geq l(n)$ such that there are no solutions to
\[a_1=a_2+\cdots+a_r\]
with $a_i\in B$ all distinct.

Estimate $l(n)$. In particular, is it true that $l(n)n^{-1/2}\to \infty$? Is it true that $l(n)< n^{1-c}$ for some $c>0$?

Erdős observed that $l(n)\geq (n/2)^{1/2}$, which Choi improved to $l(n)>(1+c)n^{1/2}$ for some $c>0$. Erdős thought he could prove $l(n)=o(n)$ but had 'difficulties in reconstructing [his] proof'.

OPEN

Let $g(n)$ be minimal such that there exists $A\subseteq \{0,\ldots,n\}$ of size $g(n)$ with $\{0,\ldots,n\}\subseteq A+A$. Estimate $g(n)$. In particular is it true that $g(n)\sim 2n^{1/2}$?

A problem of Rohrbach, who proved
\[(2^{1/2}+c)n^{1/2} \leq g(n) \leq 2n^{1/2}\]
for some small constant $c>0$.

OPEN

Let $f(n)$ be maximal such that in any $A\subset \mathbb{Z}$ with $\lvert A\rvert=n$ there exists some sum-free subset $B\subseteq A$ with $\lvert B\rvert \geq f(n)$, so that there are no solutions to
\[a+b=c\]
with $a,b,c\in B$. Estimate $f(n)$.

SOLVED

Let $A\subseteq \{1,\ldots,n\}$ with $\lvert A\rvert \leq n^{1/2}$. Must there exist some $B\subset\mathbb{Z}$ with $\lvert B\rvert=o(n^{1/2})$ such that $A\subseteq B+B$?

A problem of Erdős and Newman [ErNe77], who proved that there exist $A$ with $\lvert A\rvert\asymp n^{1/2}$ such that if $A\subseteq B+B$ then
\[\lvert B\rvert \gg \frac{\log\log n}{\log n}n^{1/2}.\]

Resolved by Alon, Bukh, and Sudakov [ABS09], who proved that for any $A\subseteq \{1,\ldots,n\}$ with $\lvert A\rvert \leq n^{1/2}$ there exists some $B$ such that $A\subseteq B+B$ and \[\lvert B\rvert \ll \frac{\log\log n}{\log n}n^{1/2}.\]

See also [333].

SOLVED

Let $c,\epsilon>0$ and $n$ be sufficiently large. If $A\subset \mathbb{N}$ has $\lvert A\rvert=n$ and $G$ is any graph on $A$ with at least $n^{1+c}$ edges then
\[\max(\lvert A+_GA\rvert,\lvert A\cdot_G A\rvert) \geq \lvert A\rvert^{1+c-\epsilon},\]
where
\[A+_GA = \{ a+b : (a,b)\in G\}\]
and similarly for $A\cdot_GA$.

A problem of Erdős and Szemerédi, which strengthens the conjecture [52].

This strong conjecture was disproved by Alon, Ruzsa, and Solymosi [ARS20], who constructed (for arbitrarily large $n$) a set of integers $A$ with $\lvert A\rvert=n$ and a graph $G$ with $\gg n^{5/3-o(1)}$ many edges such that \[\max(\lvert A+_GA\rvert,\lvert A\cdot_G A\rvert) \ll \lvert A\rvert^{4/3+o(1)}.\] Alon, Ruzsa, and Solymosi do prove, however, that if $A$ has size $n$ and $G$ has $m$ edges then \[\max(\lvert A+_GA\rvert,\lvert A\cdot_G A\rvert) \gg m^{3/2}n^{-7/4}.\]

OPEN

Let $k\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\{1,\ldots,N\}$ contains some $A$ of size $\lvert A\rvert=n$ such that
\[\langle A\rangle = \left\{\sum_{a\in A}\epsilon_aa: \epsilon_a\in \{0,1\}\right\}\]
contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that
\[g_3(n) \gg 3^n?\]

A problem of Erdős and Sárközy who proved
\[g_3(n) \gg \frac{3^n}{n^{O(1)}}.\]

SOLVED

Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$. Is it true that
\[\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{(\log \lvert A\rvert)^C}\]
for some constant $C>0$?

OPEN

Let $f(N)$ be maximal such that there exists $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=\lfloor N^{1/2}\rfloor$ such that $\lvert (A+A)\cap [1,N]\rvert=f(N)$. Estimate $f(N)$.

Erdős and Freud [ErFr91] proved
\[\left(\frac{3}{8}-o(1)\right)N \leq f(N) \leq \left(\frac{1}{2}+o(1)\right)N,\]
and note that it is closely connected to the size of the largest quasi-Sidon set (see [840]).

OPEN

Let $f(N)$ be the size of the largest quasi-Sidon subset $A\subset\{1,\ldots,N\}$, where we say that $A$ is quasi-Sidon if
\[\lvert A+A\rvert=(1+o(1))\binom{\lvert A\rvert}{2}.\]
How does $f(N)$ grow?

Considered by Erdős and Freud [ErFr91], who proved
\[\left(\frac{2}{\sqrt{3}}+o(1)\right)N^{1/2} \leq f(N) \leq \left(2+o(1)\right)N^{1/2}.\]
Note that $2/\sqrt{3}=1.15\cdots$. The lower bound is taking a genuine Sidon set $B\subset [1,N/3]$ of size $\sim N^{1/2}/\sqrt{3}$ and taking the union with $\{N-b : b\in B\}$. The upper bound was improved by Pikhurko [Pi06] to
\[f(N) \leq \left(\left(\frac{1}{4}+\frac{1}{(\pi+2)^2}\right)^{-1/2}+o(1)\right)N^{1/2}\]
(the constant here is $=1.863\cdots$).

The analogous question with $A-A$ in place of $A+A$ is simpler, and there the maximal size is $\sim N^{1/2}$, as proved by Cilleruelo.

OPEN

Let $A\subset \mathbb{N}$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there is a subset of size at least $\epsilon n$ which contains no three-term arithmetic progression.

Is it true that $A$ is the union of a finite number of sets which contain no three-term arithmetic progression?