Asks for which graphs $G$ every balanced coloring of a large $K_n$ with $e(G)$ colors, in which each vertex sees equally many edges of every color, contains a rainbow copy of $G$.

• Graph theory
• Ramsey theory

Asks whether consecutive diagonal Ramsey numbers satisfy $R(n+1)/R(n)\ge1+c$ for a constant $c>0$, and whether $R(n+1)-R(n)\gg n^2$.

• Graph theory
• Ramsey theory

Estimates the largest clique forced in an $n$-vertex graph in which every seven vertices span a triangle. The exponent lies between $5/12$ (Bucić and Sudakov) and $1/2$ (Erdős and Hajnal), and whether either bound can be improved is open.

• Graph theory

Estimates $f(N)$, the largest size of $(A+A)\cap[1,N]$ for $A\subseteq\{1,\ldots,N\}$ with $\lfloor N^{1/2}\rfloor$ elements; open. Erdős and Freud (1991) bounded $f(N)/N$ asymptotically by $3/8$ and $1/2$, and an unreviewed 2026 note claims the lower constant $0.469$.

• Additive combinatorics

Asks whether $\gcd(2^n-1,3^n-1)=1$ for infinitely many $n$, and how fast the least bases $k<l$ with $\gcd(k^n-1,l^n-1)=1$ grow.

• Number theory

Asks whether the number $h(x)$ of coprime pairs $a<b<x$ with $\sigma(a)=\sigma(b)$ satisfies $h(x)>x^{2-o(1)}$. An AI-assisted write-up of August 2026 claims $h(x)>x^{\gamma}$ for every $\gamma<13/8$, and a draft of October 2026 claims the answer yes under a hypothesis on primes $p$ with $p+1$ free of large prime factors.

• Number theory

Asks whether infinitely many $n$ have $\tau(n+k)\ll k$ for every $k\ge1$, where $\tau$ is the number of divisors.

• Number theory

Asks how many points in general position in the plane force $k$ of them whose triples all determine circles of distinct radii.

• Geometry

Asks whether, for every integer $a$, there are infinitely many $n$ with $\phi(n)\mid n+a$.

• Number theory

Asks whether the number of ways to write $n$ as a sum of two cubes is at most $(\log n)^{O(1)}$.

• Number theory

Asks whether there are infinitely many amicable pairs, and whether the number $A(x)$ of them up to $x$ satisfies $A(x)>x^{1-o(1)}$.

• Number theory

Asks to estimate the least number of distinct radii among circles through three of $n$ points in the plane, with no three collinear and no four concyclic.

• Geometry

Asks whether, for some $k>2$, the $k$-element subsets of $\{1,\ldots,2k\}$ can be colored with $k+1$ colors so that the $k$-subsets of every $(k+1)$-set show all $k+1$ colors.

• Graph theory
• Hypergraphs

Asks for the set $A_3$ of $\alpha$ such that $3$-uniform hypergraphs of limiting density above $\alpha$ must contain growing subgraphs of density above some fixed $\beta>\alpha$, while limiting density at least $\alpha$ need not.

• Graph theory
• Hypergraphs

Asks to estimate the least number $f(n)$ of distinct convex subsets determined by $n$ points in the plane with no three collinear, and in particular whether $\log f(n)/(\log n)^2$ tends to a constant.

• Geometry
• Convexity

Asks whether an increasing sequence of positive integers in which no term is a sum of consecutive earlier terms must satisfy $\limsup a_n/n=\infty$.

• Number theory

Asks how fast the largest quasi-Sidon subset of $\{1,\ldots,N\}$ grows, where $A$ is quasi-Sidon if $\lvert A+A\rvert=(1+o(1))\binom{\lvert A\rvert}{2}$.

• Additive combinatorics
• Sidon sets

Asks whether, for every $t\ge1$, some integer $a$ has exactly $t$ solutions of $\binom{n}{k}=a$ with $1\le k\le n/2$.

• Number theory
• Binomial coefficients

Asks whether there are distinct integers $x$ and $y$ such that $x$ and $y$, $x+1$ and $y+1$, and $x+2$ and $y+2$ each have the same prime factors.

• Number theory
• Primes

Asks to estimate $h(x)$, the length of the longest run of pairwise distinct consecutive prime gaps $d_n,d_{n+1},\ldots$ starting at some $n<x$. In particular, it asks whether $h(x)>(\log x)^c$ for some $c>0$ and whether $h(x)=o(\log x)$.

• Number theory
• Primes

Asks whether $r(x)$, the least even number that is not a prime gap $p_{n+1}-p_n$ with $n\le x$, tends to infinity, and whether even $r(x)/\log x\to\infty$.

• Number theory
• Primes

Asks, for the integers $a_1<a_2<\cdots$ coprime to the $k$th primorial $n_k$ (the product of the first $k$ primes), for an estimate of the smallest even number not of the form $a_{i+1}-a_i$, and whether $\gg\max_i(a_{i+1}-a_i)$ even numbers have that form.

• Number theory

Asks whether the prime-counting function satisfies $\pi(x+y)\le\pi(x)+\pi(y)$ for all large $x$ and $y$.

• Number theory
• Primes

Asks, for $k\ge3$, for an estimate of the largest $\sum_{n\in A}1/n$ over sets $A\subseteq\{1,\ldots,N\}$ with no $k$ members sharing the same pairwise least common multiple.

• Number theory

Asks for an estimate of the least $m$ such that any $m$ subsets of $\{1,\ldots,n\}$ contain a sunflower of size $k$, meaning $k$ of the sets with equal pairwise intersections.

• Combinatorics