Asks whether $2^{n-2}+1$ points in the plane, no three on a line, always contain the vertices of a convex $n$-gon.

• Geometry
• Convexity

Asks to determine $k(n,m)$, the fewest vertices forcing every directed graph to contain an independent set of size $n$ or a transitive tournament of size $m$. It is open, with exact values known when $n=1$ or $m\le2$, when $n=2$ and $m\le6$, and when $m=3$ and $n\le5$.

• Graph theory
• Ramsey theory

Asks whether, among monic complex polynomials $p$ of degree $n$, the length of the curve $\{z:|p(z)|=1\}$ is maximized by $z^n-1$.

• Polynomials
• Analysis

Asks whether, for every infinite set $A$ of reals, some set of positive measure contains no affine copy $aA+b$ with $a\neq0$.

• Combinatorics

Asks for which number theoretic functions $f$ the shifted values $n+f(n)$ cluster unboundedly in short intervals infinitely often.

• Number theory

Asks whether, for integers $3\le d_1<\cdots<d_r$ with $\sum_i 1/(d_i-1)\ge1$, every large integer is a sum of one number from each $P(d_i,0)$, the set of sums of distinct powers of $d_i$.

• Number theory
• Base representations
• Complete sequences

Asks whether a graph on $n$ vertices must contain a triangle if every induced subgraph on at least $\lfloor n/2\rfloor$ vertices has more than $n^2/50$ edges.

• Graph theory

Asks how large the chromatic and clique numbers can be for the graph joining points at integer distance in an infinite plane set with no three points on a line and no four on a circle.

• Graph theory
• Chromatic number

Asks whether any $n\ge5$ points in the plane determine two distances that each occur between at most $n$ pairs, and whether the number of such distances tends to infinity with $n$.

• Distances

Asks whether the product of $k\ge3$ consecutive positive integers can ever be powerful, meaning that $p^2$ divides it whenever a prime $p$ does.

• Number theory
• Powerful numbers

Asks whether, for every $k\ge3$, there are $k$ consecutive primes in arithmetic progression.

• Additive combinatorics
• Primes
• Arithmetic progressions

Asks for an asymptotic formula for $r_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ with no non-trivial $k$-term arithmetic progression.

• Additive combinatorics
• Arithmetic progressions

Asks whether the average of the $\alpha$-th powers of the gaps between consecutive squarefree numbers up to $x$ has a limit for every $\alpha\geq0$.

• Number theory

Asks for good estimates of $F(k)$, the number of ways to write $1$ as a sum of reciprocals of $k$ distinct positive integers.

• Number theory
• Unit fractions

Asks whether the strong chromatic index of every graph, the least number of induced matchings that partition its edges, is at most $\frac54\Delta^2$, where $\Delta$ is the maximum degree. It is open, and the best refereed bound is $1.772\Delta^2$ for large $\Delta$.

• Graph theory

Asks whether every graph on $n$ vertices has a clique transversal of at most $n-H(n)$ vertices (Erdős–Gallai), where $H(n)$ is the independence bound for triangle-free graphs. It is open; the best bounds are $n-\sqrt{2n}+\sqrt2$ explicitly and $n-c\sqrt{n\log n}$ asymptotically.

• Graph theory

Asks whether, for a finite Sidon set $A$, the mean squared gap between consecutive elements of $A+A$ tends to infinity as $\lvert A\rvert\to\infty$.

• Sidon sets

Asks whether the largest size $F(N)$ of a Sidon subset of $\{1,\ldots,N\}$ satisfies $F(N+k)\leq F(N)+1$ for every fixed $k\geq 1$ and all large $N$.

• Additive combinatorics
• Sidon sets

Asks whether $\{1,\ldots,N\}$ contains a maximal Sidon set of size $O(N^{1/3})$.

• Sidon sets

Asks whether every infinite $A\subset\mathbb{N}$ in which each $n$ has at most two representations $a+b=n$ with $a\leq b$ must have $\liminf \lvert A\cap\{1,\ldots,N\}\rvert/N^{1/2}=0$.

• Sidon sets

Asks whether $R(C_4,K_n)\ll n^{2-c}$ for some constant $c>0$.

• Graph theory
• Ramsey theory

Asks for estimates of $h(N)$, the least number of colors for $\{1,\ldots,N\}$ such that every four-term arithmetic progression gets at least three distinct colors.

• Additive combinatorics
• Arithmetic progressions

Asks whether the least $m$ such that some 2-coloring of the complete $t$-uniform hypergraph on $n$ vertices gives every set of $m$ or more vertices at least an $\alpha$ fraction of edges in each color varies continuously as $\alpha$ goes from 0 to 1/2, or jumps.

• Combinatorics
• Hypergraphs
• Ramsey theory
• Discrepancy

Asks whether the size threshold beyond which some 2-coloring of $K_n$ balances every large induced subgraph grows like $c_\alpha\log n$. With the threshold corrected to the smallest such size, this is the question of Problem 563, which is open.

• Combinatorics
• Ramsey theory
• Discrepancy

Asks for an asymptotic formula for $R(3,k)$. The order $k^2/\log k$ is known, and the constant lies between $1/2$ and $1$.

• Graph theory
• Ramsey theory