Asks whether, under the generalized continuum hypothesis, every set mapping on $\omega_{\omega+1}$ whose values have size at most $\aleph_\omega$ and pairwise intersections of size below $\aleph_\omega$ has a free set of size $\aleph_{\omega+1}$.

• Set theory
• Combinatorics

Asks whether some $K_4$-free graph forces a monochromatic triangle, and whether some $K_{\aleph_1}$-free graph forces a monochromatic $K_{\aleph_0}$, in every edge coloring with countably many colors.

• Set theory
• Ramsey theory

Asks whether, for each uncountable cardinal $\kappa$, there is a cardinal $\lambda$ such that every graph of chromatic number $\lambda$ contains a triangle-free subgraph of chromatic number $\kappa$.

• Set theory
• Chromatic number

Asks whether every graph of chromatic number $\aleph_1$ has an edge coloring with $\aleph_1$ colors such that every countable coloring of its vertices has a color class containing edges of all colors.

• Set theory
• Ramsey theory

Asks to prove that $d_r(e)=(r-2)e+3$ for all $r,e\ge3$, where $d_r(e)$ is the least $d$ such that forbidding all $r$-uniform hypergraphs with $d$ vertices and $e$ edges forces $o(n^2)$ edges.

• Graph theory
• Hypergraphs

Asks whether, for some $c>0$ and all large $n$, the least prime not dividing $\prod_{1\le i\le\log n}(n+i)$ is below $(1-c)(\log n)^2$. Only the trivial bound $(1+o(1))(\log n)^2$ is known.

• Number theory

Asks whether, for $n=k^{\alpha+o(1)}$ with $\alpha>1$, the number of integers among $n+1,\ldots,n+k$ whose largest prime factor exceeds $k$ is $(1-\rho(\alpha)+o(1))k$, with $\rho$ the Dickman function.

• Number theory
• Primes

Estimates $\delta_k$, where every $2$-coloring of $\{1,\ldots,n\}$ has at least $(\delta_k+o(1))n^2$ monochromatic $k$-term arithmetic progressions.

• Additive combinatorics
• Arithmetic progressions

Asks whether every infinite Sidon set $A$ has $\lvert A\cap[1,x]\rvert=o\bigl((x/\log x)^{1/2}\bigr)$ along a subsequence, and whether some infinite Sidon set has $\lvert A\cap[1,x]\rvert\gg x^{1/2}/(\log x)^c$ for some $c>0$.

• Additive combinatorics
• Sidon sets

Asks whether, for every $r\ge2$, some basis $A$ of order $r$ has $\sum_{n\le x}f_r(n)^2\ll x$, where $f_r(n)$ counts the representations of $n$ as a sum of $r$ elements of $A$.

• Additive combinatorics
• Additive bases

Asks how fast $a_n/n$ must grow when $A\subset\mathbb{N}$ is a set in which every integer $n\ge1$ is uniquely a difference $a_n-b_n$ of two elements of $A$.

• Additive combinatorics
• Additive bases
• Sidon sets

Asks for a constant $C$ such that, for every large $x$, there are primes $p_1<\cdots<p_k<x$ with $\sum 1/p_i<C$ and congruences $a_i\pmod{p_i}$ that cover every integer $n<x$.

• Number theory
• Primes

Asks whether $F(n)=\max_k\omega(n+k)\log\log k/\log k$ tends to infinity as $n\to\infty$, where $\omega$ counts distinct prime factors.

• Number theory

Asks for estimates of the least possible largest element $A(k)$ and least possible mean $B(k)$ of $k$ integers missing a residue class modulo every prime; open. Asymptotically, $A(k)$ lies between $\tfrac12k\log k$ and $k\log k$, and $B(k)$ between $\tfrac14k\log k$ and $\tfrac12k\log k$.

• Number theory

Asks whether $\{1,2^3,\ldots,N^3\}$ contains a Sidon set of size $\gg N$, and whether the cubes of some set of positive density form a Sidon set. A Sidon set has all pairwise sums distinct.

• Number theory
• Sidon sets

Asks for an estimate of $F_d(n)$, the largest number of points with all distances distinct that every set of $n$ points in $\mathbb{R}^d$ must contain, for fixed $d\ge2$ as $n\to\infty$.

• Geometry
• Distances

Asks whether one $n$ making every $n+a_k$ prime, for a fast-growing integer sequence $a_k$, forces infinitely many such $n$, with squarefree and $n+2^{2^k}$ variants. Three of the questions are answered no and three remain open.

• Number theory

Asks whether every set $A$ of pairwise coprime integers in $[1,n)$ satisfies $\sum_{a\in A}\frac{1}{n-a}\le\sum_{p<n}\frac{1}{p}+O(1)$.

• Number theory

Asks whether, under GCH, the partition relation for triples $\aleph_{\beta+1}\to(\aleph_\beta,4,\ldots,4)^3$ fails for $\beta=\omega_{\omega+1}$, with countably infinitely many targets equal to $4$.

• Set theory
• Ramsey theory