Unsolved Problems

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...

Find the optimal constant $c>0$ such that the following holds. For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partition into two ...

Does there exist $B\subset\mathbb{N}$ which is not an additive basis, but is such that for every set $A\subseteq\mathbb{N}$ of Schnirelmann density $\...

Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $ \lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon} $ for all $\epsilon>0$...

For what functions $g(N)\to \infty$ is it true that $ \lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)} $ implies $\limsup 1_A\ast 1_A(n)=\in...

Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial coincidences)....

Let $M\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\subset \{1,\ldots,N\}$ there is another Sidon set ...

If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that $ \binom{\lvert A\rvert}{2}+\binom{\lvert B\r...

Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\ldots,M\}$ (wh...

Schoenberg proved that for every $c\in [0,1]$ the density of $ \{ n\in \mathbb{N} : \phi(n)<cn\} $ exists. Let this density be denoted by $f(c)$. Is i...

Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the sma...

Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ $ \max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\e...

Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?...

For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either ...

If $G_1,G_2$ are two graphs with chromatic number $\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\aleph_0$) whic...

Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots $ be the lengths of cycles in $G$. Is it true that $ \sum\frac{1}{a_i}\gg \lo...

Is there $A\subseteq \mathbb{N}$ such that $ \lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n} $ exists and is $ eq 0$?...

Is $ \sum_{n\geq 2}\frac{1}{n!-1} $ irrational?...

Let $\mathfrak{c}$ be the ordinal of the real numbers, $\beta$ be any countable ordinal, and $2\leq n<\omega$. Is it true that $\mathfrak{c}\to (\beta...

Let $f(n)\to \infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made...

Is there a graph of chromatic number $\aleph_1$ such that for all $\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then...

If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, ...

Give a constructive proof that $R(k)>C^k$ for some constant $C>1$....

Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in a...

Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $...

Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\log n\t...

The cycle set of a graph $G$ on $n$ vertices is a set $A\subseteq \{3,\ldots,n\}$ such that there is a cycle in $G$ of length $\ell$ if and only if $\...

Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$ with...

Let $\epsilon >0$. Is it true that, if $k$ is sufficiently large, then $ R(G)>(1-\epsilon)^kR(k) $ for every graph $G$ with chromatic number $\chi(G)=...

Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances?...

Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(1/\log\log n)}$ many pairs which are distance 1 apart?...

Let $n$ be a sufficently large integer. Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between ...

Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^2$ in which every $x\in A$ has at least $f(n)$ points in $A$ equid...

If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart....

Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distance...

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...