Asks whether, for each graph $H$, there is $c=c(H)>0$ such that every graph on $n$ vertices with no induced copy of $H$ has a clique or an independent set of size at least $n^c$.

• Graph theory

Asks whether any two graphs of chromatic number $\aleph_1$ must share a common subgraph of chromatic number $4$, or even $\aleph_0$.

• Graph theory

Asks whether every finite graph with minimum degree at least $3$ contains a cycle of length $2^k$ for some $k\ge2$.

• Graph theory
• Cycles

Asks whether the reciprocals of the distinct cycle lengths of a graph with $n$ vertices and $kn$ edges sum to $\gg\log k$, and whether a complete bipartite graph minimizes that sum.

• Graph theory
• Cycles

Asks whether some $A\subseteq\mathbb{N}$ has its number of representations $1_A\ast 1_A(n)$, divided by $\log n$, tending to a nonzero limit.

• Number theory
• Additive bases

Asks whether $\sum_{n\ge2}\frac{1}{n!-1}$ is irrational.

• Number theory
• Irrationality

Asks whether the order type $\mathfrak{c}$ of the real line satisfies $\mathfrak{c}\to(\beta,n)_2^3$ for every countable ordinal $\beta$ and every finite $n\ge2$.

• Graph theory
• Ramsey theory
• Set theory

Asks whether some graph with $\aleph_1$ vertices and chromatic number $\aleph_1$ has, for every $\epsilon>0$ and all large $n$, an independent set of size $>n^{1-\epsilon}$ in each $n$-vertex subgraph, and asks the same for size $\gg n$.

• Graph theory
• Chromatic number

Asks for the value of $\lim_{k\to\infty}R(k)^{1/k}$, where $R(k)$ is the Ramsey number of $K_k$; the limit is not known to exist.

• Graph theory
• Ramsey theory

Asks for a constructive proof that $R(k)>C^k$ for some constant $C>1$, or equivalently for an explicit graph on $n$ vertices with clique and independence numbers $O(\log n)$.

• Graph theory
• Ramsey theory

Asks to estimate $f_c(n)$, the largest book (an edge in many triangles) forced in an $n$-vertex graph with at least $cn^2$ edges, each in a triangle. It is open: the question whether $f_c(n)>n^{\epsilon}$ for some $\epsilon>0$ was answered no, and the question whether $f_c(n)\gg\log n$ is open.

• Graph theory
• Ramsey theory

Asks whether $F(n)/\log n\to\infty$, where $F(n)$ is the size of the largest regular induced subgraph guaranteed in every graph on $n$ vertices.

• Graph theory

Asks whether the number $f(n)$ of cycle sets of graphs on $n$ vertices is $o(2^n)$, and whether $f(n)/2^{n/2}\to\infty$.

• Graph theory
• Cycles

Asks whether the least minimum degree $f(n)$ forcing a $C_4$ in an $n$-vertex graph satisfies $f(n+1)\ge f(n)$ for all large $n$. It is open, and the equivalent star Ramsey numbers $R(C_4,K_{1,n})$ are known exactly only for small $n$ and near prime-power squares.

• Graph theory
• Ramsey theory

Asks whether every subgraph of the $n$-dimensional hypercube with at least $(1/2+o(1))n2^{n-1}$ edges, slightly more than half of all its edges, must contain a $C_4$.

• Graph theory

Asks whether, for each $0<\epsilon<1$ and all large $k$, every graph $G$ with $\chi(G)=k$ has $R(G)>(1-\epsilon)^kR(k)$, or even $R(G)>cR(k)$ for a fixed $c>0$. Erdős's original conjecture, $R(G)\ge R(k)$, fails at $k=4$.

• Graph theory
• Ramsey theory

Asks whether every set of $n$ distinct points in the plane determines $\gg n/\sqrt{\log n}$ distinct distances.

• Geometry
• Distances

Asks whether, for large $n$, there are at least two non-similar $n$-point sets in the plane that minimize the number of distinct distances.

• Geometry
• Distances

Asks whether the least number $h(n)$ of distinct distances among $n$ points in the plane, with no three on a line and no four on a circle, satisfies $h(n)/n\to\infty$.

• Geometry
• Distances

Asks whether, for all large $n$, an $n$-point set in the plane with minimum distance $1$ and the smallest possible diameter must contain three points forming an equilateral triangle of side $1$.

• Geometry
• Distances

Asks whether $n$ points in the plane whose pairwise distances are at least $1$, and whose distinct distances differ by at least $1$, must have diameter $\gg n$.

• Geometry
• Distances

Asks whether $n$ points in the plane, no five on a line, determine only $o(n^2)$ lines containing exactly four of them.

• Geometry

Asks how many collinear points are forced when $n$ points in the plane determine at least $cn^2$ lines with more than three points each, and whether this number $h_c(n)$ tends to infinity for fixed $c>0$.

• Geometry

Asks whether the number of incongruent $n$-point sets in the plane with minimum distance $1$ and the least possible diameter tends to infinity.

• Geometry
• Distances

Asks whether $n$ points in the plane lie on only $o(n^2)$ distinct unit circles that each contain at least three of them.

• Geometry