Asks for an estimate of the least $n\ge2k$ such that $n-i$ divides $\binom{n}{k}$ for all but one $0\le i<k$.

• Number theory

Asks whether infinitely many primes have the form $2^kq+1$ with $q$ prime, or the form $2^k3^lq+1$.

• Number theory

Asks to estimate $g(n)$, the largest $g$ such that every graph on $n$ points in the plane at pairwise distance at least $1$, with edges joining the pairs at distance exactly $1$, has an independent set of size $g$.

• Graph theory
• Planar graphs

Asks whether every graph of chromatic number $\aleph_1$ contains a countable subgraph that is infinitely vertex-connected.

• Graph theory
• Set theory
• Chromatic number

Asks, for $f(p)$ the least $n$ with $p\mid n!+1$, whether $f(p)=p-1$ for infinitely many primes $p$ and whether $f(p)/p\to 0$ for almost all $p$.

• Number theory

Asks whether the number $A(x)$ of composite $u<x$ dividing $n!+1$ for some $n$ satisfies $A(x)\le x^{o(1)}$.

• Number theory

Asks whether the set of $m$ such that $m!+1$ has a prime factor $p\not\equiv1\pmod m$, and the set of primes arising this way (Pillai primes, as a proportion of all primes), have densities, and what they are.

• Number theory

Asks whether $n$ points in the plane with no three collinear always determine at least $\lfloor n/2\rfloor$ distinct distances, even as seen from a single point.

• Geometry
• Distances

Asks to estimate the largest number of pairs at distance exactly $1$ among $n$ points in $\mathbb{R}^d$ that are pairwise at distance at least $1$.

• Geometry
• Distances

Asks to estimate the largest number of pairs at distance exactly $1$ among $n$ points in $\mathbb{R}^d$.

• Geometry
• Distances

Asks to estimate the largest number of triangles of equal area whose vertices come from a set of $n$ points in the plane.

• Geometry
• Distances

Asks to estimate the largest number of four-point subsets with two pairs at equal distance among $n$ points in the plane, and whether it is at most $n^{3+o(1)}$.

• Geometry
• Distances

Asks for estimates of the least $f_d(n)$ such that any $f_d(n)$ points in $\mathbb{R}^d$ contain $n$ points with all pairwise distances distinct, and whether $f_d(n)=2^{o(d)}$ for fixed $n\ge3$.

• Geometry

Asks whether infinitely many $\binom{n}{k}$ with $n\ge 2k$ have deficiency $1$, and whether only finitely many have deficiency $>1$. The deficiency, defined when no prime $p\le k$ divides $\binom{n}{k}$, counts those of $n,\ldots,n-k+1$ whose prime factors are all at most $k$.

• Number theory
• Binomial coefficients

Asks whether, for all $n\ge 2k$ with only finitely many exceptions, the least prime factor of $\binom{n}{k}$ is at most $\max(n/k,k)$.

• Number theory
• Binomial coefficients

Asks for an estimate of $g(k)$, the smallest $n>k+1$ for which every prime factor of $\binom{n}{k}$ exceeds $k$.

• Number theory
• Binomial coefficients

Asks how many distinct values can occur as the common difference of a three-term arithmetic progression inside a set of $n$ integers.

• Number theory
• Additive combinatorics

Asks whether the number $\tau_\perp(n)$ of coprime consecutive divisor pairs of $n$ satisfies $\tau_\perp(n)/\omega(n)\to\infty$ for almost all $n$ and $\tau_\perp(n)<\exp((\log n)^{o(1)})$ for all $n$, and how its maximum over squarefree $n$ with $\omega(n)=k$ grows.

• Number theory
• Divisors

Asks whether a good sequence of pairwise coprime integers with convergent reciprocal sum can have $u_n<n^{O(1)}$, or at least $u_n\le e^{o(n)}$.

• Number theory

Asks how fast an infinite set of integers must grow if every sum of two of its members is squarefree.

• Number theory

Asks for an estimate of the largest chromatic number of a triangle-free graph on $n$ vertices.

• Graph theory
• Chromatic number

Asks whether the number $F(n)$ of distinct prime factors of $p(1)p(2)\cdots p(n)$, with $p$ the partition function, tends to infinity, and whether $F(n)>n$ for all large $n$.

• Number theory

Asks whether, for $r\geq 2$, every large integer is the sum of at most $r+1$ numbers divisible by the $r$th power of each of their prime factors.

• Number theory
• Powerful numbers

Asks whether the set of sums of distinct factorials contains only finitely many $k$th powers for each $k\ge 2$, and only finitely many powerful numbers.

• Number theory
• Factorials

Asks for an estimate of the size $f(N)$ of the largest subset of $\{1,\ldots,N\}$ whose pairwise sums are all squarefree, and whether $f(N)\le N^{o(1)}$.

• Number theory