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

Is every odd $n$ the sum of a squarefree number and a power of 2?

Odlyzko has checked this up to $10^7$. Granville and Soundararajan [GrSo98] have proved that this is very related to the problem of finding primes $p$ for which $2^p\equiv 2\pmod{p^2}$ (for example this conjecture implies there are infinitely many such $p$).

This is equivalent to asking whether every $n$ not divisible by $4$ is the sum of a squarefree number and a power of two. Erdős thought that proving this with two powers of 2 is perhaps easy, and could prove that it is true (with a single power of two) for almost all $n$.

SOLVED - $500

If $G$ is an edge-disjoint union of $n$ copies of $K_n$ then is $\chi(G)=n$?

Conjectured by Faber, Lovász, and Erdős (apparently 'at a party in Boulder, Colarado in September 1972' [Er81]).

Kahn [Ka92] proved that $\chi(G)\leq (1+o(1))n$ (for which Erdős gave him a 'consolation prize' of \$100). Hindman has proved the conjecture for $n<10$. Kang, Kelly, Kühn, Methuku, and Osthus [KKKMO21] have proved the answer is yes for all sufficiently large $n$.

In [Er97d] Erdős asks how large $\chi(G)$ can be if instead of asking for the copies of $K_n$ to be edge disjoint we only ask for their intersections to be triangle free, or to contain at most one edge.

SOLVED - $500

Let $f(n)$ be minimal such that there is an intersecting family $\mathcal{F}$ of sets of size $n$ (so $A\cap B\neq\emptyset$ for all $A,B\in \mathcal{F}$) with $\lvert \mathcal{F}\rvert=f(n)$ such that any set $S$ with $\lvert S\rvert \leq n-1$ is disjoint from at least one $A\in \mathcal{F}$.

Is it true that \[f(n) \ll n?\]

Conjectured by Erdős and Lovász [ErLo75], who proved that
\[\frac{8}{3}n-3\leq f(n) \ll n^{3/2}\log n\]
for all $n$. The upper bound was improved by Kahn [Ka92b] to
\[f(n) \ll n\log n.\]
(The upper bound constructions in both cases are formed by taking a random set of lines from a projective plane of order $n-1$, assuming $n-1$ is a prime power.)

This problem was solved by Kahn [Ka94] who proved the upper bound $f(n) \ll n$. The Erdős-Lovász lower bound of $\frac{8}{3}n-O(1)$ has not been improved, and it has been speculated (see e.g. [Ka94]) that the correct answer is $3n+O(1)$.

In [Er97f] Erdős asks about $f_\epsilon(n)$, defined analogously except with $\lvert S\rvert \leq n-1$ replaced by $\lvert S\rvert \leq (1-\epsilon)n$. He asks whether $f_\epsilon(n)/n\to \infty$ as $\epsilon \to 0$.

OPEN

Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?

The blow-up of $C_5$ shows that this would be the best possible. The best known bound is due to Balogh, Clemen, and Lidicky [BCL21], who proved that deleting at most $1.064n^2$ edges suffices.

SOLVED

Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?

Győri proved this with $1.03n^5$, which has been improved by Füredi. The answer is yes, as proved independently by Grzesik [Gr12] and Hatami, Hladky, Král, Norine, and Razborov [HHKNR13].

In [Er97f] Erdős asks more generally: if $r\geq 5$ is odd and a graph has $rn$ vertices and the smallest odd cycle has size $r$ then is the number of cycles of size $r$ at most $n^{r}$?

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

For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\geq n^c$ vertices?

OPEN - $500

Is there $A\subseteq \mathbb{N}$ such that
\[\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}\]
exists and is $\neq 0$?

A suitably constructed random set has this property if we are allowed to ignore an exceptional set of density zero. The challenge is obtaining this with no exceptional set. Erdős believed the answer should be no. Erdős and Sárkzözy proved that
\[\frac{\lvert 1_A\ast 1_A(n)-\log n\rvert}{\sqrt{\log n}}\to 0\]
is impossible. Erdős suggests it may even be true that the $\liminf$ and $\limsup$ of $1_A\ast 1_A(n)/\log n$ are always separated by some absolute constant.

OPEN - $500

Let $f(n)\to \infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges?

Conjectured by Erdős, Hajnal, and Szemerédi [EHS82].

Rödl [Ro82] has proved this for hypergraphs, and also proved there is such a graph (with chromatic number $\aleph_0$) if $f(n)=\epsilon n$ for any fixed constant $\epsilon>0$.

It is open even for $f(n)=\sqrt{n}$. Erdős offered \$500 for a proof but only \$250 for a counterexample. This fails (even with $f(n)\gg n$) if the graph has chromatic number $\aleph_1$ (see [111]).

OPEN - $100

Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$ with
\[\geq \left(\frac{1}{2}+o(1)\right)n2^{n-1}\]
many edges contains a $C_4$?

The best known result is due to Balogh, Hu, Lidicky, and Liu [BHLL14], who proved that $0.6068 n2^{n-1}$ edges suffice.

A similar question can be asked for other even cycles.

See also [666] and the entry 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].

SOLVED

Suppose $n$ points in $\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then
\[\sum_i f(u_i)^2 \ll n^3.\]

SOLVED - $500

Let $x_1,\ldots,x_n\in\mathbb{R}^2$ determine the set of distances $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then for all $\epsilon>0$
\[\sum_i f(u_i)^2 \ll_\epsilon n^{3+\epsilon}.\]

OPEN

If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.

Conjectured by Erdős and Moser. Füredi [Fu90] proved an upper bound of $O(n\log n)$. A short proof of this bound was given by Brass and Pach [BrPa01]. The best known upper bound is
\[\leq n\log_2n+4n,\]
due to Aggarwal [Ag15].

Edelsbrunner and Hajnal [EdHa91] have constructed $n$ such points with $2n-7$ pairs distance $1$ apart. (This disproved an early stronger conjecture of Erdős and Moser, that the true answer was $\frac{5}{3}n+O(1)$.)

OPEN

Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\gg n$?

Perhaps the diameter is even $\geq n-1$ for sufficiently large $n$. Piepmeyer has an example of $9$ such points with diameter $<5$. Kanold proved the diameter is $\geq n^{3/4}$. The bounds on the distinct distance problem [89] proved by Guth and Katz [GuKa15] imply a lower bound of $\gg n/\log n$.

OPEN

Is there some $F(n)$ such that every graph with chromatic number $\aleph_1$ has, for all large $n$, a subgraph with chromatic number $n$ on at most $F(n)$ vertices?

Conjectured by Erdős, Hajnal, and Szemerédi [EHS82]. This fails if the graph has chromatic number $\aleph_0$.

OPEN

If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges.

What is the behaviour of $h_G(n)$? Is it true that $h_G(n)/n\to \infty$ for every graph $G$ with chromatic number $\aleph_1$?

A problem of Erdős, Hajnal, and Szemerédi [EHS82]. Every $G$ with chromatic number $\aleph_1$ must have $h_G(n)\gg n$ since $G$ must contain, for some $r$, $\aleph_1$ many vertex disjoint odd cycles of length $2r+1$.

On the other hand, Erdős, Hajnal, and Szemerédi proved that there is a $G$ with chromatic number $\aleph_1$ such that $h_G(n)\ll n^{3/2}$. In [Er81] Erdős conjectures that this can be improved to $\ll n^{1+\epsilon}$ for every $\epsilon>0$.

See also [74].

OPEN

Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.

Estimate $h(n)$ as well as possible.

SOLVED

Let $\alpha$ be a cardinal or ordinal number or an order type such that every two-colouring of $K_\alpha$ contains either a red $K_\alpha$ or a blue $K_3$. For every $n\geq 3$ must every two-colouring of $K_\alpha$ contain either a red $K_\alpha$ or a blue $K_n$?

Conjectured by Erdős and Hajnal. In arrow notation, this is asking where $\alpha \to (\alpha,3)^2$ implies $\alpha \to (\alpha, n)^2$ for every finite $n$.

The answer is no, as independently shown by Schipperus [Sc99] (published in [Sc10]) and Darby [Da99].

For example, Larson [La00] has shown that this is false when $\alpha=\omega^{\omega^2}$ and $n=5$. There is more background and proof sketches in Chapter 2.9 of [HST10], by Hajnal and Larson.

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 - $100

Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\in\mathbb{R}$ and $a\neq 0$?

The Erdős similarity problem.

This is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\{a_1>a_2>\cdots\}$ is a countable strictly monotone sequence which converges to $0$.

Steinhaus [St20] has proved this is false whenever $A$ is a finite set.

This conjecture is known in many special cases (but, for example, it is is open when $A=\{1,1/2,1/4,\ldots\}$. For an overview of progress we recommend a nice survey by Svetic [Sv00] on this problem.

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

Are there, for all large $n$, some points $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is
\[o\left(\frac{n}{\sqrt{\log n}}\right)?\]

One can also ask this for points in $\mathbb{R}^3$. In $\mathbb{R}^4$ Lenz observed that there are $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^4$ such that $d(x_i,y_j)=1$ for all $i,j$, taking the points on two orthogonal circles.

More generally, if $F(2n)$ is the minimal number of such distances, and $f(2n)$ is minimal number of distinct distances between any $2n$ points in $\mathbb{R}^2$, then is $f \ll F$?

See also [89].

OPEN

Let $c<1$ be some constant and $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ be such that $\lvert A_i\rvert >c\sqrt{n}$ for all $i$ and $\lvert A_i\cap A_j\rvert\leq 1$ for all $i\neq j$.

Must there exist some set $B$ such that $B\cap A_i\neq \emptyset$ and $\lvert B\cap A_i\rvert \ll_c 1$ for all $i$?

This would imply in particular that in a finite geometry there is always a blocking set which meets every line in $O(1)$ many points.

In [Er81] the condition $\lvert A_i\cap A_j\rvert\leq 1$ for all $i\neq j$ is replaced by every two points in $\{1,\ldots,n\}$ being contained in exactly one $A_i$, that is, $A_1,\ldots,A_m$ is a pairwise balanced block design (and the condition $c<1$ is omitted).

OPEN

Is there some constant $c$ such that for every $n$ there are $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that $\lvert A_i\rvert >n^{1/2}-c$ for all $i$, and $\lvert A_i\cap A_j\rvert \leq 1$ for all $i\neq j$, and every pair $1\leq x<y\leq n$ has $\{x,y\}\subseteq A_i$ for some $i$?

A problem of Erdős and Larson [ErLa82].

Shrikhande and Singhi [ShSi85] have proved that the answer is no conditional on the conjecture that the order of every projective plane is a prime power, by proving that every pairwise balanced design on $n$ points in which each block is of size $\geq n^{1/2}-c$ can be embedded in a projective plane of order $n+i$ for some $i\leq c+2$, if $n$ is sufficiently large.

Erdős asks if this is false for constant, for which functions $h(n)$ will the condition $\lvert A_i\rvert \geq n^{1/2}-h(n)$ make the conjecture true?

SOLVED

Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that, for every $\epsilon>0$, if $n$ is sufficiently large, every subgraph of $Q_n$ with
\[\geq \epsilon n2^{n-1}\]
many edges contains a $C_6$?

OPEN

Let $p,q\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices.
Is
\[c(p,q)=\liminf \frac{\log H(n)}{\log n}\]
a strictly increasing function of $q$ for $1\leq q\leq \binom{p-1}{2}+1$?

A problem of Erdős, Faudree, Rousseau, and Schelp.

When $q=1$ this corresponds exactly to the classical Ramsey problem, and hence for example \[\frac{1}{p-1}\leq c(p,1) \leq \frac{2}{p+1}.\] It is easy to see that if $q=\binom{p-1}{2}+1$ then $c(p,q)=1$. Erdős, Faudree, Rousseau, and Schelp have shown that $c(p,\binom{p-1}{2})\leq 1/2$.

OPEN

Let $F_k(n)$ be minimal such that for any $n$ points in $\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points.

Estimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\lim F_k(n)/n^2$ and $\lim f_k(n)/n^2$.

Trivially $f_k(n)\leq F_K(n)$ and $f_2(n)=F_2(n)=\binom{n}{2}$. The problem with $k=3$ is the classical 'Orchard problem' of Sylvester. Burr, Grünbaum, and Sloane [BGS74] have proved that
\[f_3(n)=\frac{n^2}{6}-O(n)\]
and
\[F_3(n)=\frac{n^2}{6}-O(n).\]
There is a trivial upper bound of $F_k(n) \leq \binom{n}{2}/\binom{k}{2}$, and hence
\[\lim F_k(n)/n^2 \leq \frac{1}{k(k-1)}.\]

OPEN

Given $a_{i}^n\in [-1,1]$ for all $1\leq i\leq n<\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ and $p_{i}^n(a_{i'}^n)=0$ if $1\leq i'\leq n$ with $i\neq i'$. We similarly define
\[\mathcal{L}^nf(x) = \sum_{1\leq i\leq n}f(a_i^n)p_i^n(x),\]
the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i^n$ for $1\leq i\leq n$.

Is there such a sequence of $a_i^n$ such that for every continuous $f:[-1,1]\to \mathbb{R}$ there exists some $x\in [-1,1]$ where \[\limsup_{n\to \infty} \sum_{1\leq i\leq n}\lvert p_{i}^n(x)\rvert=\infty\] and yet \[\mathcal{L}^nf(x) \to f(x)?\]

Bernstein [Be31] proved that for any choice of $a_i^n$ there exists $x_0\in [-1,1]$ such that
\[\limsup_{n\to \infty} \sum_{1\leq i\leq n}\lvert p_{i}^n(x)\rvert=\infty.\]
Erdős and Vértesi [ErVe80] proved that for any choice of $a_i^n$ there exists a continuous $f:[-1,1]\to \mathbb{R}$ such that
\[\limsup_{n\to \infty} \lvert \mathcal{L}^nf(x)\rvert=\infty\]
for almost all $x\in [-1,1]$.