Unsolved Problems

Define Ulam's sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, \ldots$ where $u_1 = 1, u_2 = 2$, and $u_{n+1}$ is the smallest number uniquely ...

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

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

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 $G$ 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...

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 $\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?...

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?...

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

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?...

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 $[N]^d$ 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 ...

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...

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)....

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

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

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?...

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

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 $100\sqrt{p}$...

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 $A$...

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...

How many rotated (about the origin) copies of the "pyjama set" $\{(x, y) \in \mathbb{R}^2 : \operatorname{dist}(x, \mathbb{Z}) \leq \varepsilon\}$ are...

Let $N$ be large. For each prime $p$ with $N^{0.51} \leq p < 2N^{0.51}$, pick a residue $a(p) \in \mathbb{Z}/p\mathbb{Z}$. Is $\#\{n \in [N] : n \equi...

Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leq p_1 < p_2 < \dots < p_{1000} < N^{9/10}$. Does the remaining set have s...

Can we pick residue classes $a_p \pmod p$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least 10 of them?...

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

If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions?...

Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that $ \lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C? $ ...

Let $A$ be the set of all odd integers not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?...

Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of 2?...

Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with $ \liminf \frac...

Let $A\subseteq \mathbb{N}$. Let $B\subseteq \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements ...

Is it true that $ \sum_{n=1}^\infty(-1)^n\frac{n}{p_n} $ converges, where $p_n$ is the sequence of primes?...

Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\leq p...

We call $m$ practical if every integer $n<m$ is the sum of distinct divisors of $m$. If $m$ is practical then let $h(m)$ be such that $h(m)$ many divi...

Let $f(n,k)$ be minimal such that every family $\mathcal{F}$ of $n$-uniform sets with $\lvert \mathcal{F}\rvert \geq f(n,k)$ contains a $k$-sunflower....

Let $n_1<n_2<\cdots$ be an arbitrary sequence of integers, each with an associated residue class $a_i\pmod{n_i}$. Let $A$ be the set of integers $n$ s...

If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$....

Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$, $ h(N) = N^{1/2}+O_\epsilon(N^\epsilon)? $...

Is there a set $A\subset\mathbb{N}$ such that $ \lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2) $ and such that every large integer can be written as...

Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest possible val...