857 problems

For $n\ge 2$, let $S(n)$ be the $n$th Schur number. Let $L(n)$ be the least length such that every sequence $A$ in $\mathbb N_+$ with $|A|=L(n)$ and average $\mu(A)\le n$ satisfies…

For a finite set $A\subseteq\mathbb{Z}_{>0}$, define its subset-sum set by $H(A)=\left\{\sum_{a\in B}a:B\subseteq A\right\}$. Let $g_3(n)$ be the least integer $N$ such that there…

Let $s\geq 3$ be an integer. An $s$-spanning set of size $2$ in a finite abelian group is a set of two elements whose signed sums of at most $s$ terms cover the group. Largest nonc…

For every fixed integer $d\ge 1$, there exists a constant $C_d>0$ such that the following holds. If $F:\mathbb{R}^n\to\mathbb{R}$ is a degree-$d$ multilinear polynomial containing…

For every integer $n\ge 6$ and every integer $k$ with $3\le k\le n-3$, there exists a $k$-dimensional $\mathbb{F}_2$-subspace $E\subseteq\mathbb{F}_{2^n}$ such that…

For a fixed integer $k\ge 2$, let $\Gamma$ be a finite abelian group of order $n$ with no nonzero element whose order divides $k$. Define $f_k(\Gamma)$ to be the largest integer su…

Let $G$ be a compact connected abelian group. Is there a sufficiently small-measure regime in which every compact set $A\subseteq G$ satisfying $\mu_G(A+A)<3\mu_G(A)$ is contained…

For each fixed integer $k\ge 2$, define $\alpha_k$ as the supremum of the real numbers $a$ such that, for almost every positive integer $n$, there exist $k$ distinct divisors of…

For every nontrivial finite abelian group $G$, let $\mathsf d(G)$ denote the maximum length of a zero-sum-free sequence over $G$. Let $\nu(G)$ be the least integer $\ell$ such that…

For every function $f$ in the broad class of non-polynomial functions considered by Bergelson, Moreira, and Richter, every $ell\textbackslash in cmathbb{N}$, every invertible measu…

For every positive integer $n$, let $[n]=\{1,\ldots,n\}$ and let $\mathcal{F}\subseteq 2^{[n]}$ be a family such that for every $A,B\in\mathcal{F}$, the intersection $A\cap B$ cont…

For every pair of integers $f\ge 1$ and $g>f$, define the increasing sequence $S_{f,g}=(a_n)_{n\ge 1}$ by $a_1=f$, $a_2=g$, and, recursively, $a_{n+1}$ is the smallest integer grea…

There exists a constant $\varepsilon>0$ such that, for every measurable function $f:\mathbb{R}\to\mathbb{R}^{+}$ for which the relevant Gowers norms are finite and nonzero,…

Let $p$ be prime, and let $A,B\subseteq\mathbb{Z}/p\mathbb{Z}$ be nonempty subsets with $|A|\ne|B|$. Define the restricted sumset…

Determine whether the following sharp upper-density bound holds: if $A\subseteq\mathbb{N}$ satisfies $ab+1$ is not squarefree for every $a,b\in A$, then…

For every integer $d\ge 2$, every prime $p$, every $H\in[1,p)$ satisfying $\ln\ln p=o(\ln H)$, and every coefficient vector $a=(a_1,\ldots,a_d)\in(\mathbb Z/p\mathbb Z)^d$, one has…

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For a finite set $A\subset\mathbb Z$, let $M_{\{0,1,3\}}(A)$ be the number of pairs $(x,d)\in\mathbb Z\times(\mathbb Z\setminus\{0\})$ such that $\{x,x+d,x+3d\}\subseteq A$. Determ…

For integers $n$ and $u$ with $u=n^{1+o(1)}$, and a prime $p>u$, define…

For every integer $n \ge 0$, every real number $K \ge 1$, and every nonempty set $A \subseteq \mathbb{F}_2^n$ satisfying $\lvert A+A\rvert \le K\lvert A\rvert$, there exist a subsp…

For every finite nonabelian group $G$, the Gao constant satisfies $E(G)=\mathsf d(G)+|G|$, where $\mathsf d(G)$ is the largest integer $\ell$ for which there exists a product-one-f…

For every fixed integer $d\geq 1$, let $[n]^d=\{1,\ldots,n\}^d$, let $M([n]^d)$ denote the maximum cardinality of a sum-free subset of $[n]^d$, and let $N_d(n)$ be the number of su…

Fix an integer $t\geq 3$. For a $t$-element set $A\subset\mathbb{Z}$, let $\Delta(A)=\{a_j-a_i:a_i,a_j\in A,\ a_i<a_j\}$ be its set of positive differences. Call $A$ a Golomb ruler…

For every prime power $q$ and integer $k\ge 0$, let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb{F}_q$ whose associated codimension-$k$ partia…

For every finite nonabelian group $G$, let $\exp(G)$ be the exponent of $G$. Define $s(G)$ to be the least integer $\ell$ such that every sequence over $G$ of length at least…