Unsolved Problems

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 $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that $ R(Q_n) \ll 2^n. $ ...

Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determin...

Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges....

Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common ...

What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic ...

Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\{1,\ldots,N\}$ (into any number of colours) there is always either a monochromat...

Let $S\subseteq \mathbb{Z}^3$ be a finite set and let $A=\{a_1,a_2,\ldots,\}\subset \mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\in S$ ...

What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\cdots<x_k$?...

Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression? In other words, given a permutation $x$ of $\mathbb{N}$ must ...

Can $\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?...

Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$?...

Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression...

Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\equiv a_i\pmod...

Is there an integer $m\geq 1$ with $(m,6)=1$ such that none of $2^k3^\ell m+1$ are prime, for any $k,\ell\geq 0$?...

Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\epsilon>0$ and large $n$, $ s_{n+1}-s_n \ll_\epsilon s_n^{\epsi...

Is there a dense subset of $\mathbb{R}^2$ such that all pairwise distances are rational?...

Let $n\geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?...

For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that...

Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infin...

Let $n_1<n_2<\cdots$ be the sequence of integers which are the sum of two squares. Explore the behaviour of (i.e. find good upper and lower bounds for...

Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that $ \sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2. $ ...

For every $c\geq 0$ the density $f(c)$ of integers for which $ \frac{p_{n+1}-p_n}{\log n}< c $ exists and is a continuous function of $c$. ", "dif...

Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Is it true that $f(n)=o(\log n)$?...

Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such that the dif...

Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from the trivial co...

Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that $ \lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1 $ and $\sum\frac{1}{a_n}\in \mathbb{Q}$. Th...

Let $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?...

Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that $ \limsup \frac{a_n}{n}=\infty. $ Is $ \sum_{n=1}^\infty \frac{1}{2^{a_n}} $ transcende...

Is $ \sum_n \frac{\phi(n)}{2^n} $ irrational? Here $\phi$ is the Euler totient function....

Is $ \sum \frac{p_n}{2^n} $ irrational? (Here $p_n$ is the $n$th prime.)...

Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is $ \sum \frac{\sigma_k(n)}{n!} $ irrational?...

Let $A\subseteq \mathbb{N}$ be such that $ \lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty $ and $ \sum_{n\in A...

Let $n\geq 1$ and $f(n)$ be maximal such that for any integers $1\leq a_1\leq \cdots \leq a_n$ we have $ \max_{\lvert z\rvert=1}\left\lvert \prod_{i}(...

Let $A\subseteq \mathbb{N}$ be an infinite set. Is $ \sum_{n\in A}\frac{1}{2^n-1} $ irrational?...

Let $a_1,a_2,\ldots$ be a sequence of positive integers with $a_n\to \infty$. Is $ \sum_{n} \frac{\tau(n)}{a_1\cdots a_n} $ irrational, where $\tau(n)...

Let $a_1<a_2<\cdots$ be an increasing sequence such that $a_n/n\to \infty$. Is the sum $ \sum_n \frac{a_n}{2^{a_n}} $ irrational?...

Are there infinitely many $n$ such that there exists some $t\geq 2$ and distinct integers $a_1,\ldots,a_t\geq 1$ such that $ \frac{n}{2^n}=\sum_{1\leq...

Let $a_n$ be a sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\to 1$ the sum $ \sum\frac{1}{b_n} $...

Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n eq 0$ and $b_n eq 0$ for all $n$) t...

Let $1\leq a_1<a_2<\cdots$ be an increasing sequence of integers. How fast can $a_n\to \infty$ grow if $ \sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\f...

Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1<n_2<\cdots $ be an infinite sequence with $n_{k+1}/n_k \geq c>1$. Must $...

Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in...

Let $A(n)=\{a_0<a_1<\cdots\}$ be the sequence defined by $a_0=0$ and $a_1=n$, and for $k\geq 1$ define $a_{k+1}$ as the least positive integer such th...

Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic progress...

Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\geq 5$?...

If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly...

Is there an infinite Lucas sequence $a_0,a_1,\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$ such that all $a_k$ are composite, and yet no integer h...

Let $A=\{n_1<\cdots<n_r\}$ be a finite set of positive integers. What is the maximum density of integers covered by a suitable choice of congruences $...

Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $...