Asks whether the order of every finite projective plane must be a prime power.

• Combinatorics

Asks whether the largest number of mutually orthogonal Latin squares of order $n$ is $\gg n^{1/2}$.

• Combinatorics

Asks for an asymptotic formula for the number of $k\times n$ Latin rectangles.

• Combinatorics

Asks whether $\sum1/p$ over the primes $p\le n$ for which $n\bmod p$ lies in $(p/2,p)$ is asymptotic to $\tfrac12\log\log n$.

• Number theory

Asks whether, for each $k\ge2$, $(n+k)!^2\mid(2n)!$ for infinitely many $n$.

• Number theory
• Factorials

Asks for a reasonable $f(n)$ such that, for almost all $n$, the least $m$ with $m\nmid\binom{2n}{n}$ satisfies $m\sim f(n)$.

• Number theory
• Binomial coefficients

Asks, for all large $n$, for a non-trivial pairwise balanced design on $n$ points in which each block size is used only $O(n^{1/2})$ times.

• Combinatorics

Asks whether, for every graph $G$ of chromatic number $\aleph_1$ and every cardinal $m$, there is a graph of chromatic number $m$ all of whose finite subgraphs occur in $G$.

• Graph theory
• Chromatic number

Asks whether every triangle-free graph of infinite chromatic number contains every tree as an induced subgraph.

• Graph theory
• Chromatic number

Asks whether a graph of infinite chromatic number $\mathfrak{m}$ must have a subgraph of chromatic number $\mathfrak{n}$ for every infinite cardinal $\mathfrak{n}<\mathfrak{m}$.

• Graph theory
• Chromatic number

Asks whether $K_n$ can always be written as the edge-disjoint union of given trees $T_2,\ldots,T_n$, where $T_k$ has $k$ vertices. This tree packing conjecture of Gyárfás is open, and one arXiv proof claim was withdrawn.

• Graph theory

Asks whether, for each $\epsilon>0$, some $A\subseteq\mathbb{N}$ has $A+A$ of lower density at least $1-\epsilon$ while every integer has $O_\epsilon(1)$ representations as a sum of two elements of $A$.

• Additive combinatorics

Asks for the best constant $c$ such that $n$ reals whose every four-element subset $B$ has $\lvert B-B\rvert\ge11$ contain a Sidon set of size $cn$.

• Geometry
• Distances
• Sidon sets

Asks whether large chromatic number forces large dichromatic number, and whether large cochromatic number forces a subgraph of large dichromatic number. Cochromatic classes induce complete or empty graphs, and dichromatic colorings avoid monochromatic oriented cycles.

• Graph theory
• Chromatic number

Asks for estimates of $f(n;k,l)$, the least Turán number over graphs with $k$ vertices and $l$ edges, for $k<l\le k^2/4$, and whether it is strictly monotone in $l$. Open; asymptotics are known only at the pairs $(5,6)$ and $(6,9)$.

• Graph theory
• Turán numbers

Asks about the densities and growth of $h(n)$, the least $h$ with $\gcd(2^n-1,3^n-1,\ldots,h^n-1)=1$.

• Number theory

Asks for the size of the largest Sidon subset of the squares $\{1,2^2,\ldots,N^2\}$, and whether it is $N^{1-o(1)}$.

• Number theory
• Sidon sets
• Squares

Asks whether every infinite set of natural numbers whose finite subsets each contain a dissociated subset (one with distinct subset sums) of proportional size is a finite union of dissociated sets.

• Number theory

Two players alternately color the edges of a complete graph, and the first wins if her largest monochromatic clique beats her opponent's. Asks whether the first player can win.

• Graph theory

Asks whether, for the product $P=p_1\cdots p_n$ of the first $n>1$ primes, there is always a prime $p$ with $p_n<p<P$ such that $P+p$ is prime.

• Number theory
• Primes

Asks whether the squares contain arbitrarily long quasi-progressions, with all gaps between $d$ and $d+C$ for a fixed $C$, and arbitrarily large cubes $a+\{\sum_i\epsilon_ib_i:\epsilon_i\in\{0,1\}\}$.

• Number theory

Asks how large a subset of any $n$ real numbers can be chosen with no two distinct elements summing to one of the $n$ numbers (the Erdős–Moser function $g(n)$). Known bounds are $(\log n)^{1+c}\ll g(n)\ll\exp(O(\sqrt{\log n}))$ for some $c>0$.

• Additive combinatorics

Estimates $f(n)$, the size of the largest sum-free subset (no solution to $a+b=c$) guaranteed inside any set of $n$ integers. The main term $n/3$ is settled and the second-order term is $o(n)$; an unrefereed 2025 preprint claims that it is at least $c\log\log n$.

• Additive combinatorics

Asks for which functions $g(n)$ there is a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices has a clique and an independent set of size $\log n$.

• Graph theory

Asks whether there is an $\epsilon>0$ such that, for all large $n$, some graph on $n$ vertices with at least $\epsilon n^2$ edges can be edge-colored with $n$ colors so that every $C_4$ gets four distinct colors.

• Graph theory
• Ramsey theory