43 problems

Let $x_1,x_2,\ldots \in (0,1)$ be an infinite sequence and let $A_k=\limsup_{n\to \infty}\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert,$ where $e(x)=e^{2\pi ix}$. Is it true tha…

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…

Call $x_1,x_2,\ldots \in (0,1)$ well-distributed if, for every $\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\subseteq [0,1]$,…

As far as we know the following related more general problem is still open. Let $A_k = \{a_1^{(k)} < a_2^{(k)} < ...\}$, $k = 1, 2, ...$ be an infinite class of infinite sets of in…

For a finite set $X\subseteq\mathbb{N}$ and a coloring $f:\mathbb{N}\times\mathbb{N}\to\{-1,1\}$, define its imbalance on $X$ by … Let $H(n)$ be the minimum, over all such coloring…

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$. Estimate the growth of, for almost all $\alpha$, $\sum_{1\leq k\leq N}f(\{ \alpha n_k\}).$ For exam…

For every measurable $E⊂(0,1)$, is it true that for almost every $α$, the proportion of $1≤k≤n$ with fractional part of $kα$ in $E$ tends to the measure of $E$?

Let $x_1<x_2<\cdots$ be an infinite sequence of integers. Is it true that, for almost all $\alpha \in [0,1]$, the discrepancy…

Choose $n$ points $w_1,…,w_n$ on the unit sphere to maximize $∏_{i<j}|w_i-w_j|$. Is their spherical-cap discrepancy $o(n)$?

For an infinite planar point sequence $A$, let $f(r)=\max_C ||A∩C|-πr^2|$, with the maximum over circles of radius $r$. Must $f(r)$ be unbounded, and how fast must it grow?

For an $n$-point set $P$ on the unit sphere, let $D(P)$ be the maximum over spherical caps $C$ of $||P∩C|-α_Cn|$, where $α_C$ is the normalized area of $C$. Does…

For every infinite sequence $x_1,x_2,…$ in $[0,1]$, must there be an interval $I$ for which \limsup_{n→∞}|#{j≤n:x_j∈I}-n|I||=∞?

Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$,…

Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced subgraph $H$ on at least $k$ v…

Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete $t$-uniform hypergraph on $n$ vertices suc…

Let $(x_1, x_2, x_3, \ldots)$ be a sequence with $x_i \in \{+1, -1\}$ for every positive integer $i$, and let $C$ be an integer. Then there exist positive integers $k$ and $d$ such…

Linear discrepancy conjecture. For every $C>0$, there is an integer $N$ such that, whenever $n\geq N$, every non-diagonal $n\times n$ matrix $M$ with entries in $\{-1,1\}$ and

For integers $r\ge 1$ and real numbers $n\ge 1$, $0<\eta\le 1/2$, define … Here $a_i,b_j\sim n$ denotes the same dyadic range as in the source, and $\lVert\cdot\rVert$ is distance…

Let $\mathcal{M}=(E,\mathcal{I})$ be a matroid whose ground set $E$ can be partitioned into $k$ disjoint bases, and let $S_1,S_2,\ldots,S_\ell$ be pairwise disjoint subsets of $E$.…

Mohr–Pardey–Rautenbach's conjecture. Fix integers $k,n\geq 1$ such that $\binom{n}{2}$ and $(k-1)n/k$ are both even integers. If $\mathbf{c}:E(K_n)\to\{-1,1\}$ is a 0-sum labeling…

Mohr–Pardey–Rautenbach conjecture. There exists a copy $F'$ of $F$ in $K_n$ such that

Far-from-Turán discrepancy conjecture. For every $\varepsilon>0$, there exists $c>0$ such that, if $G$ is an $n$-vertex graph that is $\varepsilon$-far from every Turán graph, incl…

Verstraete's conjecture.

Let $T$ be a tree with $\ell$ leaves, let $\mathcal{T}$ denote the family of trees used in the discrepancy definition, and let $\mathcal{D}^1(T,\mathcal{T})$ denote the one-dimensi…

Let $T$ be a tree with $\ell$ leaves, and let $\mathcal{DT}$ be the set of all directed rooted trees, namely trees with a distinguished root and all edges oriented away from it. Wr…