Asks whether every $r$-coloring of the edges of $K_{r^2+1}$, for $r\ge3$, has $r+1$ vertices whose induced $K_{r+1}$ misses a color.

• Graph theory

Asks to prove that $H(n)-\log_2 n\to\infty$, where $H(n)$ is the least size such that some map from the subsets of an $n$-element set $X$ to $X$ sends the subsets of every set that large onto $X$.

• Combinatorics

Asks whether, for $k\ge 4$, the largest girth of an $n$-vertex graph with chromatic number $k$, divided by $\log n$, tends to a limit.

• Graph theory
• Chromatic number
• Cycles

Asks whether $f(n)/(n/(\log_2 n)^2)$ tends to a limit, where $f(n)$ is the largest ratio $\chi(G)/\omega(G)$ of chromatic number to clique number over graphs $G$ on $n$ vertices.

• Graph theory
• Chromatic number

Asks whether a graph with chromatic number $k$ and no $K_k$ must have two disjoint subgraphs of chromatic numbers $\ge a$ and $\ge b$ whenever $a,b\ge 2$ and $a+b=k+1$.

• Graph theory
• Chromatic number

Asks for the least number of vertices $n(k)$ of a bipartite graph whose list chromatic number exceeds $k$; open, with $n(2)=6$ and $n(3)=14$ known.

• Graph theory
• Chromatic number

Asks for all $n$ such that some triangle can be cut into $n$ congruent triangles.

• Geometry

Asks how large a set $A\subseteq\{1,\ldots,N\}$ can be if $b-a\nmid b$ for all $a,b\in A$ with $b-a\ge t$.

• Number theory

Asks whether, for each $k\ge 3$, large enough chromatic number forces an odd cycle whose vertices span a subgraph of chromatic number at least $k$.

• Graph theory
• Chromatic number

Asks whether a graph on $n$ vertices in which every cycle has more vertices than chords must have at most a constant times $n$ edges.

• Graph theory
• Cycles

Asks to estimate how many edges force a $t$-uniform hypergraph on $n$ vertices to have four edges $A,B,C,D$ with $A\cup B=C\cup D$ and $A\cap B=C\cap D=\emptyset$.

• Graph theory
• Hypergraphs

Bounds the size of a set needed to meet every member of a family of $k$-element sets in which any $r$ members are all met by one pair. The question is whether this size is asymptotically a constant times $k$.

• Combinatorics

Asks whether some $n>24$ satisfies $m+\tau(m)\le n+2$ for every $m<n$, where $\tau(m)$ is the number of divisors of $m$.

• Number theory

Asks to estimate how many distinct distances to the others some point must have, among any $n$ points in the plane with no four on a circle.

• Geometry
• Distances

Asks whether $n$ planar points with no isosceles triangle must determine at least $f(n)n$ distinct distances, for some $f(n)\to\infty$.

• Geometry
• Distances

Asks whether the $n$ vertices of a convex polyhedron in $\mathbb{R}^3$ always determine at least $(1-o(1))n/2$ distinct distances.

• Geometry
• Distances
• Convexity

Asks whether, for all large $n$, two sets of $n$ planar points can have only $o(n/\sqrt{\log n})$ distinct distances between the two sets.

• Geometry
• Distances

Asks whether, for fixed $k$ and large $n$, the least prime not dividing $(n+1)\cdots(n+k)$ is less than $(1+o(1))\log n$.

• Number theory

Asks whether, for some constant $C$ and all large $n$, there is a pairwise balanced design on $\{1,\ldots,n\}$ (a family of sets in which every pair of distinct elements lies in exactly one set) whose sets all have more than $n^{1/2}-C$ elements.

• Combinatorics

Asks whether $c(p,q)$, the growth exponent of the largest clique forced when every $p$ vertices span at least $q$ edges, is strictly increasing in $q$. Open; the 1997 source gives bounds at the endpoints and a disputed upper bound near the top.

• Graph theory
• Ramsey theory

Asks whether the number of incongruent $n$-point planar sets maximizing the number of unit distances tends to infinity, and whether it exceeds $1$ for every $n>3$.

• Geometry
• Distances

Asks for bounds on how many lines can pass through at least $k$, or through exactly $k$, of $n$ points in the plane.

• Geometry

Asks whether $n$ points in $\mathbb{R}^d$ whose pairwise distances all differ by at least $1$ must have diameter at least $(1+o(1))n^2$.

• Geometry
• Distances

Asks whether the product $n(n+d)\cdots(n+(k-1)d)$ of $k\ge4$ positive terms of an arithmetic progression with $(n,d)=1$ can be a perfect power.

• Number theory

Asks which sets of integers, such as the sums of two squares, have the translation property: for every $n$, some shift $t_n\ge1$ leaves membership of each $a\le n$ unchanged.

• Number theory