Unsolved Problems

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

Find reasonable bounds for the maximal density of a set $A \subset \{1, \ldots, N\}$ not containing a 3-term progression with common difference a squa...

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...

Define the 2-colour van der Waerden numbers $W(k, r)$ to be the least quantities such that if $\{1, \dots, W(k, r)\}$ is coloured red and blue then th...

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...

What is $C$, the infimum of all exponents $c$ for which the following is true, uniformly for $0 < \alpha < 1$? Suppose that $A \subset \mathbb{F}_2^n$...

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

Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?...

Suppose that a small sieve process leaves a set of maximal size. What is the structure of that set?...

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 $K \subset \mathbb{R}^N$ be a balanced compact set with normalized Gaussian measure $\gamma_\infty(K) \geq 0.99$. Does $10K$ contain a compact con...

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....

Do $\Phi(G)$ and $\Phi'(G)$ coincide?...

Suppose $A, B \subset \{1, \dots, N\}$ both have size $N^{0.49}$. Does $A + B$ contain a composite number?...

Is every $n \leq N$ the sum of two integers, all of whose prime factors are at most $N^\varepsilon$?...

Is there an absolute constant $c > 0$ such that if $A \subset \mathbb{N}$ is a set of squares of size at least 2, then $|A + A| \geq |A|^{1+c}$?...

Suppose $A + A$ contains the first $n$ squares. Is $|A| \geq n^{1-o(1)}$?...

Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \in \{1, \dots, p-1\}$ congruent to some product $a_1a_2$ mo...

Let $A$ be the smallest set containing 2 and 3, and closed under the operation $a_1a_2 - 1$ (if $a_1, a_2 \in A$, then $a_1a_2 - 1 \in A$). Does $A$ h...

Do there exist infinitely many primes $p$ for which $p-2$ has an odd number of prime factors (counting multiplicity)?...

Is there $c > 0$ such that whenever $A \subset [N]$ has size $N^{1-c}$, the difference set $A - A$ contains a nonzero square?...

Is there always a sum of two squares between $X - \frac{1}{10}X^{1/4}$ and $X$?...

Determine bounds for Waring's problem over finite fields....

Suppose $A \subset \mathbb{F}_p^2$ is a set meeting every line in at most 2 points. Is it true that all except $o(p)$ points of $A$ lie on a cubic cur...

Fix $k$. Let $A \subset \mathbb{R}^2$ be a set of $n$ points with no more than $k$ on any line. Suppose at least $\delta n^2$ pairs $(x, y) \in A \tim...

Fix integers $k, \ell$. Given $n \geq n_0(k, \ell)$ points in $\mathbb{R}^2$, is there either a line containing $k$ of them, or $\ell$ of them that ar...

Suppose $A \subset \mathbb{R}^2$ is a set of size $n$ with $cn^2$ collinear 4-tuples. Does it contain 5 points on a line?...

What is the largest subset of the grid $[N]^2$ with no three points on a line? In particular, for $N$ sufficiently large, is it impossible to have a s...

Let $\Gamma$ be a smooth codimension 2 surface in $\mathbb{R}^n$. Must $\Gamma$ intersect some 2-dimensional plane in 5 points, if $n$ is sufficiently...

What is the largest subset of $[N]^d$ with no 5 points on a 2-plane?...

Let $X \subset \mathbb{R}^2$ be a set of $n$ points. Does there exist a line $\ell$ through at least two points of $X$ such that the numbers of points...

Let $A$ be a set of $n$ points in the plane. Can one select $A' \subset A$ of size $n/2$ such that any axis-parallel rectangle containing 1000 points ...

Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?...

Suppose $A$ is an open subset of $[0, 1]^2$ with measure $\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\geq...

Let $c > 0$ and let $A$ be a set of $n$ distinct integers. Does there exist $\theta$ such that no interval of length $\frac{1}{n}$ in $\mathbb{R}/\mat...

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...

Is a random polynomial with coefficients in $\{0, 1\}$ and nonzero constant term almost surely irreducible?...

Let $A \subset \mathbb{R}$ be a set of positive measure. Does $A$ contain an affine copy of $\{1, \frac{1}{2}, \frac{1}{4}, \dots\}$?...