Unsolved Problems

What is the minimum number of colors needed to color the points of the plane such that no two points at distance 1 have the same color?...

What is the exact value of $R(5,5)$, the smallest number $n$ such that any 2-coloring of the edges of $K_n$ contains a monochromatic $K_5$?...

For any $n$ runners on a circular track with distinct constant speeds, each runner is "lonely" (distance at least $1/n$ from all others) at some time....

For every finite union-closed family of sets (other than the empty family), there exists an element that belongs to at least half of the sets....

Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $n/3 + \Omega(n)$, where $\Omega(n) \to \infty$ as $n \to ...

Let $A \subset \mathbb{Z}$ be a set of $n$ integers. Is there a subset $S \subset A$ of size $(\log n)^{100}$ such that $S \hat{+} S$ is disjoint from...

Suppose that $A \subset [N]$ has no more than $\varepsilon N^2$ solutions to $x + y = z$. Can one remove $\varepsilon' N$ elements to leave a sum-free...

Fix an integer $d$. What is the largest sum-free subset of $[N]^d$?...

Is $r_5(N) \ll N(\log N)^{-c}$? Is $r_4(\mathbb{F}_5^n) \ll N^{1-c}$ where $N = 5^n$?...

Let $S \subset \mathbb{N}$ be random. Under what conditions is Roth's theorem for progressions of length 3 true with common differences in $S$?...

Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15}N^{10}$ tuples $(x_1, \ld...

Suppose that $A \subset \mathbb{Z}/N\mathbb{Z}$ has density $\alpha$ and is Fourier uniform (all Fourier coefficients of $1_A - \alpha$ are $o(N)$). D...

Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{Z}\} \subset \mathbb{Z}^2$ is ...

What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?...

Suppose that $A \subset \mathbb{F}_3^n$ is a set of density $\alpha$. Under what conditions on $\alpha$ is $A$ guaranteed to contain a 3-term progress...

Suppose $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Are there $\gg_\alpha |G|^3$ triples $x, y, g$ such tha...

Find reasonable bounds for instances of the multidimensional Szemerédi theorem....

Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\log(1/\alpha))...

Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?...

Suppose that $A \subset \mathbb{F}_2^n$ has an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$?...

Suppose that $\mathbb{F}_2^n$ is partitioned into sets $A_1, \dots, A_K$. Does $2A_i$ contain a coset of codimension $O_K(1)$ for some $i$?...

Let $p$ be an odd prime and suppose $f : \mathbb{F}_p^n \times \mathbb{F}_p^n \to \mathbb{C}$ is bounded pointwise by 1. Suppose $\mathbb{E}_h \|\Delt...

Determine bounds for the inverse theorem for Gowers norms....

Let $p(k)$ be the limit as $n \to \infty$ of the probability that a random permutation on $[n]$ preserves some set of size $k$. Is $p(k)$ a decreasing...

Consider a set $S \subset [N]^3$ with the property that any two distinct elements $s, s'$ of $S$ are comparable (in the coordinatewise partial order)....

Let $A \subset \mathbb{F}_2^n$. If $V$ is a subspace, write $\alpha(V)$ for the density of $A$ on $V$. Is there some $V$ of moderately small codimensi...

Suppose $A \subset \mathbb{Z}/p\mathbb{Z}$ has density $\frac{1}{2}$. Under what conditions on $K$ can $A$ be almost invariant under all maps $\phi(x)...

Given a string $x \in \{0, 1\}^n$, let $\tilde{x}$ be obtained by deleting bits independently at random with probability $\frac{1}{2}$. How many indep...

In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \times n$ chessboard?...

Suppose $a_1, \dots, a_k$ are integers which do not satisfy Rado's condition. Is $c(a_1, \dots, a_k)$ bounded in terms of $k$ only?...

If $\{1, \dots, N\}$ is $r$-coloured, then for $N \geq N_0(r)$ there exist integers $x, y \geq 3$ such that $x+y$ and $xy$ have the same colour. Find ...

If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\{0, 1, 3\}$ that $A$ can contain?...

For which values of $k$ is the following true: whenever we partition $[N] = A_1 \cup \dots \cup A_k$, we have $|\bigcup_{i=1}^k (A_i \hat{+} A_i)| \ge...

Let $A_1, \dots, A_{100}$ be "cubes" in $\mathbb{F}_3^n$ (images of $\{0, 1\}^n$ under linear automorphisms). Is $A_1 + \dots + A_{100} = \mathbb{F}_3...

What is the size of the smallest set $A \subset \mathbb{Z}/p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a ...

Suppose $X, Y$ are finitely-supported independent random variables taking integer values such that $X + Y$ is uniformly distributed on its range. Are ...

Given a set $A \subset \mathbb{Z}$ with $D(A) \leq K$, find a large structured subset $A'$ which "obviously" has $D(A') \leq K + \varepsilon$....

Write $F(N)$ for the largest Sidon subset of $[N]$. Improve, at least for infinitely many $N$, the bounds $N^{1/2} + O(1) \leq F(N) \leq N^{1/2} + N^{...

Let $p$ be a prime and let $A \subset \mathbb{Z}/p\mathbb{Z}$ be a set of size $\sqrt{p}$. Is there a dilate of $A$ with a gap of length $100\sqrt{p}$...

Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$ with $|A| = (\sqrt{2} + o(1))q^{1/2}$ and $A + A = \mathbb{Z...

Suppose $A \subset [N]$ has size $\geq c\sqrt{N}$ and representation function $r_A(n) \leq r$ for all $n$. What can be said about the structure of $A$...

For arbitrarily large $n$, does there exist an abelian group $H$ with $|H| = n^{2+o(1)}$ and subsets $A_1, \dots, A_n, B_1, \dots, B_n$ satisfying $|A...

What is the largest subset $A \subset \mathbb{F}_7^n$ for which $A - A$ intersects $\{-1, 0, 1\}^n$ only at 0?...

If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random with $|A| = \sqrt{p}$, can we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translat...

Let $r$ be fixed and let $H(r)$ be the Hamming ball of radius $r$ in $\mathbb{F}_2^n$. Let $f(r)$ be the smallest constant such that there exist infin...

In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \times n$ chessboard?...

For any finite family of finite sets that is closed under taking unions, must there exist an element that belongs to at least half of the sets?...

Does there exist a finite upper bound on how many times a number (other than 1) can appear in Pascal's triangle?...

What is the maximum size of a cap set in $\mathbb{F}_3^n$?...

Does every family of at least $c^k k!$ sets of size $k$ contain a sunflower of size 3, for some absolute constant $c$?...