Unsolved Problems

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...

Which finite groups have the smallest largest product-free sets?...

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...

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 $G$ 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...

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...

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 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....

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 $|A + A| \geq |A|^{1+c}$?...

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...

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...

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...

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

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...

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 $d \geq 3$ be odd. Give bounds on $\nu(d)$ such that if $n > \nu(d)$ then any homogeneous polynomial $F(\mathbf{x}) \in \mathbb{Z}[x_1, \dots, x_n...

Finding a single solution to $F(x_1, \dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of solutions in $A$ is roug...

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?...

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 $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \subset A$ with $|S| \gg K^{-O(1)}|A|$ and $S^8 \subset A^4$?...

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^{...

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...

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?...

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...

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings?...

What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \pmod p$, one for each prime $p \leq x$?...

Pick $x_1, \dots, x_k \in A_n$ at random. Is it true that, almost surely as $n \to \infty$, the random walk on this set of generators and their invers...

Find bounds in the classification theorem for approximate groups....

Let $d \geq 3$ be an odd integer. Give bounds on $\nu(d)$ such that if $n > \nu(d)$ the following is true: given any homogeneous polynomial $F(\mathbf...

Finding a single solution to a polynomial equation $F(x_1, \dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of su...

Is every group well-approximated by finite groups?...