Unsolved Problems

What is $\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of $3$-edges which can placed on $n$ vertices so that there exists no $K_4^3$, a set of ...

For every $x\in\mathbb{R}$ let $A_x\subset \mathbb{R}$ be a bounded set with outer measure $<1$. Must there exist an infinite independent set, that is...

What is the size of the largest $A\subseteq \mathbb{R}^d$ such that every three points from $A$ determine an isosceles triangle? That is, for any thre...

Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Es...

What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\mathbb{R}^2$ such that no two points o...

Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set $ \{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\} $ be covered by a set of ...

If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that $ \sum_{n\in A}\cos(n\theta) < ...

Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of $ \liminf_{r\to \infty} \frac{\max_n\lver...

Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$, $ \lvert f(z)/z^n\rvert \to \infty $ as $z\to \in...

Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function (with $a_k eq 0$ for all $k\geq 1$). Is it true that if $n_k/k\to \infty$ then $f(z)$ as...

Let $f$ be a Rademacher multiplicative function: a random $\{-1,0,1\}$-valued multiplicative function, where for each prime $p$ we independently choos...

Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$. If $R_n$ counts the number of real roots of $f_n(z)=\sum_{0\...

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\l...

For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for alm...

Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect...

Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\mathbb{Z}^k$ (conditional on no self intersec...

Let $\ell(N)$ be maximal such that in any finite set $A\subset \mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\ell(N)$ (i.e. the onl...

Let $F(k)$ be the minimal $N$ such that if we two-colour $\{1,\ldots,N\}$ there is a set $A$ of size $k$ such that all subset sums $\sum_{a\in S}a$ (f...

Let $\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_5$ and at least $\delta n^2$ edges then $G$ contains a set ...

Let $r\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pairwise greate...

Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be disti...

Let $r\geq 2$ and suppose that $A\subseteq\{1,\ldots,N\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and ...

Let $h(n)$ be such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set $ \left\{ \frac{a}{(a,b)}: a,b\in A\right\} $ has size at least $h(n...

Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\subseteq G$ is a random set of size $k$ ...

Show that $ R(3,k+1)-R(3,k)\to\infty $ as $k\to \infty$. Similarly, prove or disprove that $ R(3,k+1)-R(3,k)=o(k). $ ...

Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if $...

Let $m_1\leq\cdots\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex cla...

Determine the Ramsey number $ R(C_4,S_n), $ where $S_n=K_{1,n}$ is the star on $n+1$ vertices. In particular, is it true that, for any $c>0$, there ar...

Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that $ \lim_{k\to...

Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of...

Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Is it true that $ R(T;...

Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine $ R(K_{s,t};...

Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-col...

Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-col...

Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergr...

Let $F(n,\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\subseteq [n]$ with $\lvert X\r...

Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic co...

Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $ R(G...

Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges a...

Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$. Is it true that, for any $H$ with $m$ edges and...

Let $k\geq 1$. What is the best possible $c_k$ such that $ R(C_{2k+1},H)\leq c_k m $ for any graph $H$ on $m$ edges without isolated vertices?...

Show that for any rational $\alpha \in [1,2)$ there exists a bipartite graph $G$ such that $ \mathrm{ex}(n;G)\asymp n^{\alpha}. $ ...

Show that for $k\geq 3$ $ \mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}. $ ...

Is it true that $ \mathrm{ex}(n;\{C_3,C_4\})\sim (n/2)^{3/2}? $ ...

Is it true that, for $k\geq 2$, $ \mathrm{ex}(n;\{C_{2k-1},C_{2k}\})=(1+o(1))(n/2)^{1+\frac{1}{k}}. $ ...

If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have wi...

Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of $ \mathrm{ex}(n;Q...

Let $\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\delta n^2$ edges then $G$ contains ...

Let $G$ be a graph with $n$ vertices and $\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\subseteq G$ such that {UL} {LI}$H_1$ has $\gg \delta^3n^2$...

What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?...