Unsolved Problems

Let $A\subseteq\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently l...

Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they diff...

Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$....

Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than three points,...

Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\geq...

Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$....

For every $r\geq 4$ and $k\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\geq f(k,r)$ contains a subgraph of girth $...

If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges. W...

Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament o...

If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}$ maximised w...

Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x eq y$ such that $xy=yx$ can be cover...

Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$, and for $n\geq 1$ let $ p_n(z)=\prod_{i\leq n} ...

Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the sha...

For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\to 0$ for almost all $n$, there are infinitely many $x$ ...

Let $a,b,c\geq 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\ge...

For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$. Let $3\leq d_1<d_2<\cdo...

Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\{ \sum...

Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a eq b\in A}(a+b)$ has at least $f(n)$ distinct prime fa...

Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy ...

Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertice...

Let $F(N)$ be the maximal size of $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any distinct elements of $A\backslash\{a\}$. Est...

Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Mus...

We say that $N$ is powerful if whenever $p\mid N$ we also have $p^2\mid N$. Let $k\geq 3$. Can the product of any $k$ consecutive positive integers ev...

Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochromatic $k$-ter...

Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression?...

Let $r_k(N)$ be the largest possible size of a subset of $\{1,\ldots,N\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove...

Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x eq y\in A$ and integers $k\geq 1$ we have $ \lvert kx -y\rvert \geq 1. $ D...

Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha \geq 0$, $ \lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(...

If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then $ \mathrm{ex}(n;H) \ll n^{2-1/r}. ...

Let $F(k)$ be the number of solutions to $ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k}, $ where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good est...

Let $G$ be a graph with maximum degree $\Delta$. Is $G$ the union of at most $\tfrac{5}{4}\Delta^2$ sets of strongly independent edges (sets such that...

For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ on at least two vertices...

For any $M\geq 1$, if $A\subset \mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\in A+A$ such that $a+1,a-1 ot...

Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that $ \frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty $ as $\lvert A\rve...

Let $F(N)$ be the size of the largest Sidon subset of $\{1,\ldots,N\}$. Is it true that for every $k\geq 1$ we have $ F(N+k)\leq F(N)+1 $ for all suff...

Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?...

Let $A\subset \mathbb{N}$ be an infinite set such that, for any $n$, there are most $2$ solutions to $a+b=n$ with $a\leq b$. Must $ \liminf_{N\to\inft...

There exists some constant $c>0$ such that $$R(C_4,K_n) \ll n^{2-c}.$$...

Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain...

Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete $t$-uniform ...

Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced ...

Give an asymptotic formula for $R(3,k)$....

Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is $ \lim_{N\to \inft...

Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-ter...

Let $F(N)$ be the smallest possible size of $A\subset \{0,1,\ldots,N\}$ such that $\{0,1,\ldots,N\}\subset A-A$. Find the value of $ \lim_{N\to \infty...

Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements ...

In any $2$-colouring of $\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$....

A finite set $A\subset \mathbb{R}^n$ is called Ramsey if, for any $k\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\mathbb{R...

Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$ such that $ ...

Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$, $ \max_{P_d}...