Unsolved Problems

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 ...

If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that $ \sum_{n\in A}\cos(n\theta) < ...

Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of $ \liminf_{r\to \infty} \frac{\max_n\lver...

Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function (with $a_k eq 0$ for all $k\geq 1$). Is it true that if $n_k/k\to \infty$ then $f(z)$ as...

For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for alm...

Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect...

Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be disti...

Let $h(n)$ be such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set $ \left\{ \frac{a}{(a,b)}: a,b\in A\right\} $ has size at least $h(n...

Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\subseteq G$ is a random set of size $k$ ...

Let $f_k(n)$ be minimal such that if $n$ points in $\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines contai...

Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ m...

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 $\...

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

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 $A\subseteq \mathbb{R}^d$ 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)...

Give an asymptotic formula for the number of $k\times n$ Latin rectangles....

Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that, for all $t$, there are $O(n^...

Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both...

Let $\epsilon>0$. Does there exist $A\subseteq \mathbb{N}$ such that the lower density of $A+A$ is at least $1-\epsilon$ and yet $1_A\ast 1_A(n) \ll_\...

Let $r\geq 2$ and $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ be such that $A_i ot\subseteq A_j$ for all $i eq j$ and for any $t$ if there exists some $i...

Do the squares contain arbitrarily long quasi-progressions? That is, does there exist some constant $C>0$ such that, for any $k$, the squares contain ...

Let $g(n)$ be maximal such that given any set $A\subset \mathbb{R}$ with $\lvert A\rvert=n$ there exists some $B\subseteq A$ of size $\lvert B\rvert\g...

Let $f(n)$ be maximal such that if $B\subset (2n,4n)\cap \mathbb{N}$ there exists some $C\subset (n,2n)\cap \mathbb{N}$ such that $c_1+c_2 ot\in B$ fo...

Let $h(n)$ be maximal such that if $A\subseteq \mathbb{Z}$ with $\lvert A\rvert=n$ then there is $B\subseteq A$ with $\lvert B\rvert \geq h(n)$ such t...

Let $l(n)$ be maximal such that if $A\subset\mathbb{Z}$ with $\lvert A\rvert=n$ then there exists a sum-free $B\subseteq A$ with $\lvert B\rvert \geq ...

Let $g(n)$ be minimal such that there exists $A\subseteq \{0,\ldots,n\}$ of size $g(n)$ with $\{0,\ldots,n\}\subseteq A+A$. Estimate $g(n)$. In partic...

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 $k\geq 3$ 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\...