Unsolved Problems

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....

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

Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$....

Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than three points,...

Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x eq y$ such that $xy=yx$ can be cover...

Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$, and for $n\geq 1$ let $ p_n(z)=\prod_{i\leq n} ...

Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the sha...

Let $F(N)$ be the maximal size of $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any distinct elements of $A\backslash\{a\}$. Est...

Let $r_k(N)$ be the largest possible size of a subset of $\{1,\ldots,N\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove...

Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha \geq 0$, $ \lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(...

For any $M\geq 1$, if $A\subset \mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\in A+A$ such that $a+1,a-1 ot...

Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that $ \frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty $ as $\lvert A\rve...

Let $F(N)$ be the size of the largest Sidon subset of $\{1,\ldots,N\}$. Is it true that for every $k\geq 1$ we have $ F(N+k)\leq F(N)+1 $ for all suff...

Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?...

Let $A\subset \mathbb{N}$ be an infinite set such that, for any $n$, there are most $2$ solutions to $a+b=n$ with $a\leq b$. Must $ \liminf_{N\to\inft...

Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain...

Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is $ \lim_{N\to \inft...

Let $F(N)$ be the smallest possible size of $A\subset \{0,1,\ldots,N\}$ such that $\{0,1,\ldots,N\}\subset A-A$. Find the value of $ \lim_{N\to \infty...

Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$ such that $ ...

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 $a_{i+1}-a_i\in S$ ...

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

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?...

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

Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that $ \sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in...

Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with $ 1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}. $ Is i...

Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with $ 0< \left\lvert \sum_{...

Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$. Estimate $S(N)$....

What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$?...

Let $A\subset \mathbb{N}$ be an additive basis of order $2$. Must there exist $B=\{b_1<b_2<\cdots\}\subseteq A$ which is also a basis such that $ \lim...

Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a eq b$ then $a+b mid ab$. Can $A$ be 'substantially more' than the odd numbers? W...

Let $d(A)$ denote the density of $A\subseteq \mathbb{N}$. Characterise those $A,B\subseteq \mathbb{N}$ with positive density such that $ d(A+B)=d(A)+d...

Can one classify all solutions of $ \prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j) $ where $k_1,k_2>3$ and $m_1+k_1\leq m_2$? Are there...

Is it true that for every $n\geq 1$ there is a $k$ such that $ n(n+1)\cdots(n+k-1)\mid (n+k)\cdots (n+2k-1)? $ ...

Let $f(n)$ be the minimal $m$ such that $ n! = a_1\cdots a_k $ with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that $ f(n)-...

Let $ V'(x)=\#\{\phi(m) : 1\leq m\leq x\} $ and $ V(x)=\#\{\phi(m) \leq x : 1\leq m\}. $ Does $\lim V(x)/V'(x)$ exist? Is it $>1$?...

Is there a sequence $1\leq d_1<d_2<\cdots$ with density $1$ such that all products $\prod_{u\leq i\leq v}d_i$ are distinct?...

Is it true that, for all $k eq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$?...

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 $f(z)\in\mathbb{C}[z]$ 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 ...