Unsolved Problems

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...

If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by $ \delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|. $ Let $h(n...

Let $A\subset \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the nu...

Let $r,k\geq 2$ be fixed. Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such t...

Let $f(n)$ be the maximal $k$ such that in any set $A\subset \mathbb{R}$ of size $n$ there is a subset $B\subseteq A$ of size $\lvert B\rvert\geq k$ w...

Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq ...

Let $f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\ldots,z_d$ with corresponding arguments $\theta_1,...

For any $0<\alpha<1$, let $ f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}). $ Does $f(\alpha,n)$ have an asymptotic dist...

If $R(k,l)$ is the Ramsey number then prove the existence of some $c>0$ such that $ \lim_k \frac{R(k+1,k)}{R(k,k)}> 1+c. $ ...

Determine the infimum and supremum of $ \lvert \{ x\in \mathbb{R} : \lvert f(x)\rvert < 1\}\rvert $ as $f\in \mathbb{R}[x]$ ranges over all non-consta...

Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is ...

Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of $ \lvert \{ z: \lvert f(z)\rvert < 1\}\rvert, $ as $f$ ranges...

Let $f(n)$ be the size of the largest subset $A\subseteq \{1,\ldots,n\}$ such that there are no three distinct elements $a,b,c\in A$ such that $a\mid ...

Let $k\geq 2$ and define $n_k\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\binom{n}{k}$ for all but one $0\leq i<k$. Estimate $n_k$....

Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the unit square, no two of which intersect, which are maxima...

Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct distances. Est...

Let $f(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in t...

Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances ...

Let $1<q<1+\epsilon$ and consider the set of numbers of the shape $\sum_{i\in S}q^i$ (for all finite $S$), ordered by size as $0=x_1<x_2<\cdots$. Is i...

Let $ A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\textrm{ finite}\right\}. $ If $k\geq 2$, then does $A$ contain only finitely many $k$th powers...

Let $f(z)$ be an entire function which is not a monomial. Let $ u(r)$ count the number of $z$ with $\lvert z\rvert=r$ such that $\lvert f(z)\rvert=\ma...

Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let $ A=\{ n \geq 1: f(n+1)< f(n)\}. $ If $\lver...

For $x_1,\ldots,x_n\in [-1,1]$ let $ l_k(x)=\frac{\prod_{i eq k}(x-x_i)}{\prod_{i eq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$...

For $x_1,\ldots,x_n\in [-1,1]$ let $ l_k(x)=\frac{\prod_{i eq k}(x-x_i)}{\prod_{i eq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$...

For $x_1,\ldots,x_n\in [-1,1]$ let $ l_k(x)=\frac{\prod_{i eq k}(x-x_i)}{\prod_{i eq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$...