Unsolved Problems

Let $r\geq 3$. For an $r$-uniform hypergraph $G$ let $\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of vertic...

For a triangle-free graph $G$ let $h_r(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $r$ (while preserving ...

If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain?...

Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A) ot\in A$ for all $A$. Must...

Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\{A : A\subseteq X\}\to X$ so that for every $Y\subseteq X$ with $\...

The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour clas...

Let $k\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains n...

Let $\omega(G)$ denote the clique number of $G$ and $\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\chi(G)/\omega(G)$, as $G$ rang...

The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (pe...

Classify those triangles which can only be cut into a square number of congruent triangles....

Find all $n$ such that there is at least one triangle which can be cut into $n$ congruent triangles....

Let $t\geq 1$ and $A\subseteq \{1,\ldots,N\}$ be such that whenever $a,b\in A$ with $b-a\geq t$ we have $b-a mid b$. How large can $\lvert A\rvert$ be...

Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\in S$ such that if the edges of $G_n$ are coloured with $n$ colours th...

Is there some function $f$ such that for all $k\geq 3$ if a finite graph $G$ has chromatic number $\geq f(k)$ then $G$ must contain some odd cycle who...

Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than diagonals. Is it true that $f(n)\ll ...

Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges $A,B,C,D$...

Let $f(k,r)$ be minimal such that if $A_1,A_2,\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there ...

Let $f(m)$ be such that if $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert=m$ then every interval in $[1,\infty)$ of length $2N$ contains $\geq f(m)$ ...

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j eq i\}$, where the points are ordered such that $ R(x_1)\leq \cdots...

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j eq i\}$, where the points are ordered such that $ R(x_1)\leq \cdots...

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ with no four points on a circle. Must there exist some $x_i$ with at least $(1-o(1))n$ distinct distances to othe...

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least $ (1+c)\f...

Is it true that if $A\subset \mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has...

Let $x_1,\ldots,x_n\in \mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least $ (1-o(1))\frac{n}{2} $ many distinct distances betwee...

Are there, for all large $n$, some points $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is $...

Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\leq t$. For e...

Let $k\geq 2$ and $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. Is it true that, if $k$ is fixed and $n$ is suff...

A pairwise balanced design for $\{1,\ldots,n\}$ is a collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that $2\leq \lvert A_i\rvert <n$...

Let $p,q\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices...

Is it true that the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\to\...

Let $F_k(n)$ be minimal such that for any $n$ points in $\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ o...

Let $A\subseteq \mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1)...

Given $a_{i}^n\in [-1,1]$ for all $1\leq i\leq n<\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ a...

We say that $A\subset \mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\geq 1$ such that, for all $1\leq a\le...

Is every sufficiently large integer of the form $ ap^2+b $ for some prime $p$ and integer $a\geq 1$ and $0\leq b<p$?...

Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$. Is it true that for all $m\geq n+k$ $ M(n,k) eq M(m,k)? $ ...

Let $\epsilon>0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that $ \omega(n-k) < ...

Is it true that, for all sufficiently large $n$, there exists some $k$ such that $ p(n+k)>k^2+1, $ where $p(m)$ denotes the least prime factor of $m$?...

Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and $ p(n+k)>k^2, $ where $p(m)$ is the least prime factor of $m$?...

Is it true that for every $1\leq k\leq n$ the largest prime divisor of $\binom{n}{k}$, say $P(\binom{n}{k})$, satisfies $ P\left(\binom{n}{k}\right)\g...

For $0\leq k\leq n$ write $ \binom{n}{k} = uv $ where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$. Let...

Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon<k\leq n^{1-\epsilon}$ the number of distinct prime divi...

Can every integer $N\geq 2$ be written as $ N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)} $ for some $k\geq 2$ and $m\geq n+k$?...

Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\leq x$ such that every integer in $[1,y]$ ...

Define $\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\epsilon_n}<p\leq n$ such that every...

Let $n$ be sufficiently large. Is there some choice of congruence class $a_p$ for all primes $2\leq p\leq n$ such that every integer in $[1,n]$ satisf...

Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_1<p_2<\cdots$ are the primes dividing $n$ then $p_k...

Given $A\subseteq \mathbb{N}$ let $M_A=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}$ be the set of multiples of $A$. Find a necessary and sufficie...

Let $k\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\{a_1<a_2<\cdots \}$ be the set of those integers in $[n,n^k]$ which have a divis...

Let $f_{\max}(n)$ be the largest $m$ such that $\phi(m)=n$, and $f_{\min}(n)$ be the smallest such $m$, where $\phi$ is Euler's totient function. Inve...