88,106 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…

Let $f$ be the infinite Fibonacci word, let $\Sigma$ be its alphabet, and define … Here $\operatorname{Cl}(w)$ is the number of distinct closed factors of $w$, and $F_n$ denotes th…

Let $\mathbb{F}_q$ be a finite field, and let $[n,k]$ denote a linear code of length $n$ and dimension $k$ over $\mathbb{F}_q$. If $1<k<q$, then … except when $q$ is even and $k=3$…

For every finite simply connected CW complex $X$, there exists a finite discrete monoid $M$ such that …

Catalan classification conjecture. There are exactly $C_n$ families of size $2^n$ in which every two distinct paths have an even number of common edges.

For every positive integer $n\in\mathbb{Z}_{>0}$, define the Collatz map $T:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ by $T(n)=n/2$ if $n$ is even and $T(n)=3n+1$ if $n$ is odd. The conje…

Conjecture. There exist an absolute constant $c\ge 1$ and an online algorithm such that, for every finite matroid $M$ and every choice of nonnegative weights, the algorithm selects…

Problem. Is every finite uniformly dense matroid of rank $3$ cyclically orderable?

Let ${\rm UC}_X$ denote the unique closed choice operation on a represented Hausdorff space $X$, and let ${\rm C}_X$ denote closed choice on $X$. For Baire space…

Let $G$ be a graph, let $F_k(G)$ denote its $k$-token graph, and let a $k$-token reconstruction family of $F_k(G)$ be a family of subsets as defined in the source. Let…

Let $G=(X,E)$ be a locally finite Borel graph on a standard Borel space $X$. Does there always exist a Borel coloring $c:X\to\{0,1\}$ such that, for every $x\in X$, the number of n…

Let $G_W=\langle a,b,c,s,t\mid c=ab=ba,\ c^2=sas^{-1}=tbt^{-1}\rangle$. Then $G_W$ has property RD (rapid decay).

(A) Global regularity on $\mathbb{R}^3$. For every smooth, rapidly decreasing, divergence-free $u_0$, with $f\equiv 0$, there exist global smooth $u$ and $p$ satisfying the equatio…

Let $G$ be a finite simple graph with $m$ edges and clique number $\omega(G)$. Let … be the eigenvalues of the adjacency matrix of $G$. The Bollobás–Nikiforov conjecture asserts th…

Problem. Determine $\kappa$. In particular, is

Let $A$ and $B$ be poset games, with $B=2^n$ for some $n\in\mathbb{N}_0$, and suppose that $B$ is weakly-canonical. Let $A\otimes B$ be their lexicographic product, whose underlyin…

Consider the real additive white Gaussian noise sequence channel … where the noise coordinates are independent $N(0,N_0/2)$. An $(E,M,\epsilon)$ code has $M$ equiprobable messages,…

Let $P(v_1,\ldots,v_N)=\int_{\mathbb R^n}\max_i e^{-|x-v_i|^2}\,dx$ for a code consisting of $N$ vectors in $\mathbb R^n$. The simplex conjecture. For $N=n+1$ and fixed…

For every integer $n\ge 1$, energy $E>0$, and noise standard deviation $\sigma>0$, let $x_0,\ldots,x_n\in\mathbb{R}^n$ satisfy $\lVert x_i\rVert^2=E$ for all $i$, and let the signa…

Let $\operatorname{syr}$ be the Syracuse map on odd positive integers, and let $\operatorname{sft}(n)$ denote the Syracuse falling time of an odd integer $n\geq 3$, namely the leas…

For each integer $d\ge 2$, determine whether there exists a classical protocol using a finite amount of classical communication and unlimited shared randomness that exactly reprodu…

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…

Stanley's conjecture. For $n\geq4$ and $n\geq5$, respectively,

For a permutation $\sigma$ of $\{1,2,\dots,n\}$ and a permutation $\beta$ of $\{1,2,\dots,k\}$, say that $\sigma$ contains $\beta$ as a pattern if there are indices…

Problem (Propp's Problem 7). Is this condition also sufficient? In other words, is