Unsolved Problems

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\...

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...

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...

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 $f(n)$ 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 $A\subset \mathbb{R}^2$ 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 ...

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 $r\geq 2$ 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...

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_...

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...

Estimate the maximum of $F(A,B)$ as $A,B$ range over all subsets of $\{1,\ldots,N\}$, where $F(A,B)$ counts the number of $m$ such that $m=ab$ has exa...

Is there an entire non-zero function $f:\mathbb{C}\to \mathbb{C}$ such that, for any infinite sequence $n_1<n_2<\cdots$, the set $ \{ z: f^{(n_k)}(z)=...

Are $ 2^n\pm 1 $ and $ n!\pm 1 $ powerful (i.e. if $p\mid m$ then $p^2\mid m$) for only finitely many $n$?...

Let $A=\{n_1<n_2<\cdots\}$ be the sequence of powerful numbers (if $p\mid n$ then $p^2\mid n$). Are there only finitely many three-term progressions o...

Let $A$ be the set of powerful numbers (if $p\mid n$ then $p^2\mid n$). Is it true that $ 1_A\ast 1_A(n)=n^{o(1)} $ for every $n$?", "difficulty":...

Let $S\subset \mathbb{R}$ be a set containing no solutions to $a+b=c$. Must there be a set $A\subseteq \mathbb{R}\backslash S$ of cardinality continuu...