Asks to estimate the least $h(n)$ such that every interval $(m,m+h(n))$ contains distinct integers $a_1,\ldots,a_{\pi(n)}$, with $a_i$ divisible by the $i$th prime $p_i$.

• Number theory
• Primes

Asks for an estimate of the largest set $A\subseteq\{1,\ldots,N\}$ in which at most one number has more than one representation as a sum of two members.

• Number theory
• Sidon sets
• Additive combinatorics

Estimates, for $k\ge3$, how far above $N$ the size of a subset of $\{1,\ldots,2N\}$ must be to force $k$ integers whose pairwise sums all lie in the set. An unrefereed 2026 preprint claims that the threshold is $1$ for $k=3$, $3$ for $k=4$, and bounded for $k=5$.

• Number theory
• Additive combinatorics

Asks how long a game can be guaranteed to last in which two players alternately add integers from $\{2,\ldots,n\}$ to a shared set with no member dividing another, and whether it lasts at least $\epsilon n$ or $(1-\epsilon)n/2$ moves.

• Number theory
• Primitive sets

Asks whether, for every $\epsilon>0$, there is a $k$ such that fewer than $X^\epsilon$ blocks of $k$ consecutive terms of an increasing integer sequence have least common multiple below $X$.

• Number theory

Asks how slowly an infinite set of naturals can grow while its sets of sums of $r$ distinct members stay disjoint for different $r$, and for which $c$ its gaps can be at most $n^c$.

• Additive combinatorics

Compares $f(n)$, the sum over the primes $p\mid n$ of the largest power of $p$ not exceeding $n$, with $F(n)$, the largest sum of distinct pairwise coprime integers from $2$ to $n$ built from those primes.

• Number theory

Asks for an estimate of the largest sum of a set of pairwise coprime integers up to $n$, and whether, for each $k\ge2$ and large $n$, the best such set contains a number with at least $k$ prime factors.

• Number theory

Asks whether every minimal additive basis $A$ of order $k$, one that stops being a basis of order $k$ when any infinite subset is removed, has an infinite $B\subset A$ with $A\setminus B$ a basis of order $k+1$.

• Number theory
• Additive bases

Asks whether, for every $k\ge1$, there are integers $N_1<\cdots<N_k$ whose factor difference sets $D(N)=\{\lvert a-b\rvert : N=ab\}$ share at least $k$ common values.

• Number theory
• Divisors

Asks whether, for each fixed $\epsilon>0$, every large $n$ has only boundedly many divisors in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$.

• Number theory
• Divisors

Asks whether there is an absolute constant $K$ such that, for every $C>0$, every large $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+Cn^{1/4})$.

• Number theory
• Divisors

Asks whether $\max_{k\ge0}v(n,k)\to\infty$, where $v(n,k)$ counts the prime factors of $n+k$ that exceed $k$, that is, those dividing none of $n,n+1,\ldots,n+k-1$.

• Number theory

Asks whether $\liminf_n\sum_{0\le i<k}\omega_k(n+i)\le k$ for every $k\ge1$, where $\omega_k$ counts the distinct prime factors above $k$, and whether $\limsup_n\bigl(\sum_{0\le i<k}\omega(n+i)\bigr)\log\log n/\log n=1$, where $\omega$ counts all distinct prime factors.

• Number theory
• Primes

Asks whether, for each $k\ge2$ and every large enough $n$, the interval $[n,n+p_1\cdots p_k)$ contains an integer with more than $k$ distinct prime factors, where $p_i$ is the $i$th prime.

• Number theory

Asks for a necessary and sufficient condition on an increasing sequence $(b_n)$ of integers for there to be a primitive sequence $(a_n)$, in which no term divides another, with $a_n\ll b_n$.

• Number theory
• Primitive sets

Asks whether $f(2n)/f(n)$ tends to a limit, where $f(n)=\sum_{k\le n}\tau(2^k-1)$ sums the divisor counts of $2^k-1$.

• Number theory
• Divisors

Asks for an estimate of the least number of edges in an $n$-uniform hypergraph that cannot be colored with two colors but can with three.

• Combinatorics
• Hypergraphs

Estimates the least order of a tournament in which every $n$ vertices have a common dominator. Open: Erdős's 1963 upper bound and the 1965 lower bound of Szekeres and Szekeres differ by a factor of order $n$, and exact values are known only up to $n=3$.

• Graph theory

Asks whether every connected set in $\mathbb{R}^n$ has a connected subset that is neither a point nor homeomorphic to the whole set.

• Topology

Asks whether the size Ramsey number of every graph with $n$ vertices and $e\ge Cn$ edges exceeds $f(C)e$ for some $f$ growing faster than linearly. Open, with no result found beyond Erdős's 1982 statement.

• Graph theory
• Ramsey theory

Asks for a proof that the number $h(n)$ of distinct exponents in the prime factorization of $n!$ satisfies $h(n)\sim c(n/\log n)^{1/2}$ for some $c>0$.

• Number theory
• Factorials

Asks whether there are infinitely many $n$ such that the prime factorization of $n(n+1)$ has all exponents distinct.

• Number theory

Asks whether the largest number $f_k(n)$ of edges in a $k$-critical graph on $n$ vertices satisfies $f_k(n)\gg_k n^2$, whether $f_6(n)\sim n^2/4$, and whether a proposed asymptotic holds for $k\geq6$. The first is proved, the second is open, and the third fails when $k$ is not a multiple of $3$.

• Graph theory
• Chromatic number

Asks whether some graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ has every subgraph on $\aleph_1$ vertices countably colorable.

• Graph theory
• Chromatic number