Unsolved Problems

Let $k\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\lfloor n^2/4\rfloor+1$ many edges such t...

Does there exist some $\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\epsilon n^2$ ma...

Suppose $n\equiv 1\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\lfloor n/m\rfloor$ ma...

Is it true that $ \frac{R(n+1)}{R(n)}\geq 1+c $ for some constant $c>0$, for all large $n$? Is it true that $ R(n+1)-R(n) \gg n^2? $ ...

Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a cliqu...

Let $k\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\{1,\ldots,N\}$ contains some $A$ of size $\lvert A\rvert=n$ such that $ \langle A\...

Let $f(N)$ be maximal such that there exists $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=\lfloor N^{1/2}\rfloor$ such that $\lvert (A+A)\cap [1,N...

Let $H(n)$ be the smallest integer $l$ such that there exist $k<l$ with $(k^n-1,l^n-1)=1$. Is it true that $H(n)=3$ infinitely often? (That is, $(2^n-...

Let $g(n)$ count the number of $m$ such that $\phi(m)=n$. Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that $ g(n) > ...

Let $h(x)$ count the number of integers $1\leq a<b<x$ such that $(a,b)=1$ and $\sigma(a)=\sigma(b)$, where $\sigma$ is the sum of divisors function. I...

Is there an absolute constant $C>0$ such that every integer $n$ with $\sigma(n)>Cn$ is the distinct sum of proper divisors of $n$?...

Are there infinitely many $n$ such that, for all $k\geq 1$, $ \tau(n+k)\ll k? $ ...

Let $n_k$ be minimal such that if $n_k$ points in $\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\bino...

Is it true that, for any $a\in\mathbb{Z}$, there are infinitely many $n$ such that $ \phi(n) \mid n+a? $ ...

Let $A\subset\mathbb{N}$ be the set of cubes. Is it true that $ 1_A\ast 1_A(n) \ll (\log n)^{O(1)}? $ ...

We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. Are there infinitely many amicable pairs? If $A(x)$ counts the numb...

Let $h(n)$ be maximal such that in any $n$ points in $\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many c...

Let $r\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edg...

Let $k\geq 2$ and $A_k\subseteq [0,1]$ be the set of $\alpha$ such that there exists some $\beta(\alpha)>\alpha$ with the property that, if $G_1,G_2,\...

Let $f(n)$ be maximal such that any $n$ points in $\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimat...

Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that no $a_i$ is the sum of consecutive $a_j$ for $j<i$. Is it true that $ \limsup \frac{a_n...

Let $f(N)$ be the size of the largest quasi-Sidon subset $A\subset\{1,\ldots,N\}$, where we say that $A$ is quasi-Sidon if $ \lvert A+A\rvert=(1+o(1))...

Let $A\subset \mathbb{R}^2$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at...

Let $A\subset \mathbb{N}$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there is a subset of ...

Is it true that, for every integer $t\geq 1$, there is some integer $a$ such that $ \binom{n}{k}=a $ (with $1\leq k\leq n/2$) has exactly $t$ solution...

Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ al...

Let $\epsilon>0$. Is there some $r\ll_\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\geq 0$ and $n$ has at most $r$ prime...

Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n<x$ the numbers $d_n,d_{n+1},\ldots,d_{n+h(x)-1}$ ar...

Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\leq x$. I...

Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes. If $1=a_1<a_2<\cdots a_{\phi(n_k)}=n_k-1$ is the sequence of integers ...

Let $k\geq 3$ and $f_k(N)$ be the maximum value of $\sum_{n\in A}\frac{1}{n}$, where $A$ ranges over all subsets of $\{1,\ldots,N\}$ which contain no ...

Let $m=m(n,k)$ be minimal such that in any collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ there must exist a sunflower of size $k$ - that...

Let $A\subseteq \{1,\ldots,N\}$ be such that there is no solution to $at=b$ with $a,b\in A$ and the smallest prime factor of $t$ is $>a$. Estimate the...

Let $t\geq 1$ and let $d_t$ be the density of the set of integers $n\in\mathbb{N}$ for which $t$ can be represented as the sum of distinct divisors of...

Let $h(n)$ be such that, for any $m\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\leq i\leq \pi(n)$ such that $p_i\m...

Let $r\geq 2$ and let $A\subseteq \{1,\ldots,N\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\leq b$ for any...

Let $A\subseteq \{1,\ldots N\}$ be a set such that there exists at most one $n$ with more than one solution to $n=a+b$ (with $a\leq b\in A$). Estimate...

There exists a constant $C>0$ such that, for all large $N$, if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there are distinct ...

Let $k\geq 3$ and $g_k(N)$ be minimal such that if $A\subseteq \{1,\ldots,2N\}$ has $\lvert A\rvert \geq N+g_k(N)$ then there exist integers $b_1,\ldo...

If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\cup A_2$ contain a minimal additiv...

Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ then...

Consider the two-player game in which players alternately choose integers from $\{2,3,\ldots,n\}$ to be included in some set $A$ (the same set for bot...

Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that $ [a_i,a_{i+1},\ldots,a_{i+k-1}] < X, $ where the ...

Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be an infinite set such that the sets $ S_r = \{ a_1+\cdots +a_r : a_1<\cdots<a_r\in A\} $ are disjoint f...

Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be an infinite sum-free set - that is, there are no solutions to $ a=b_1+\cdots+b_r $ with $b_1<\cdots<b_...

If $n=\prod_{1\leq i\leq t} p_i^{k_i}$ is the factorisation of $n$ into distinct primes then let $ f(n)=\sum p_i^{\ell_i}, $ where $\ell_i$ is chosen ...

Call a set $S\subseteq \{1,\ldots,n\}$ admissible if $(a,b)=1$ for all $a eq b\in S$. Let $ G(n) = \max_{S\subseteq \{1,\ldots,n\}} \sum_{a\in S}a $ a...

Let $A\subset\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\subset A$ is any infinite set then $A\backslash B...

For $A\subseteq \{1,\ldots,n\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime. Is it true ...

Is it true that, for any $n$, if $d_1<\cdots <d_t$ are the divisors of $n$, then $ \sum_{1\leq i<j\leq t}\frac{1}{d_j-d_i} \ll 1+\sum_{1\leq i<t}\frac...