1,339 problems

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.

Kahn's conjecture. If $|X|=|Y|=|Z|$, then

Let $G_n$ be the partition graph, let $\mathrm{Ax}_n$ and $\mathrm{Sp}_n$ denote its axial and spinal vertex sets, and let…

Let $X$, $Y$, and $Z$ be upward closed subsets of $Q_n$. It is conjectured that … Notation: the poset $Q_n$ is the power set of a set with $n$ elements, ordered by subset relation.…

For every integer $k\ge 2$, there exists a convex function $f_k:[0,\infty)\to[0,1]$ satisfying … as $x\to\infty$, such that every uncrowded $k$-uniform hypergraph $H$ admits a loca…

For every fixed integer $d\ge 3$, there exists a constant $c_d>0$ such that, for every connected $d$-regular graph $G$ on $n$ vertices and every tree $T$ on $n$ vertices, if…

Let $k>d\ge 2$ and let $n\ge k(d+1)/d$. A $d$-simplex-cluster is a collection of $d+1$ distinct members $A_1,\dots,A_{d+1}$ of a family $\mathcal F\subseteq\binom{[n]}{k}$ such tha…

Determine all integers $r\ge 0$ for which there exists a finite simple $r$-regular graph $G$ such that the triangle-degrees are pairwise distinct: for every vertex $v\in V(G)$, let…

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…

For every regular triangle-free graph $G$ of order $n$, if $\lambda_1(G)\geq\cdots\geq\lambda_n(G)$ are the adjacency eigenvalues of $G$, then…

For a finite poset $\mathcal P$ and $n\ge 1$, define $\operatorname{sat}^{*}(n,\mathcal P)$ to be the minimum of $|\mathcal F|$ over all families $\mathcal F\subseteq 2^{[n]}$ such…

For every fixed pair of integers $r,k\ge 3$, let $C_k^{(r)}$ denote the $r$-uniform linear cycle with $k$ hyperedges, and let $\operatorname{ex}_r(n,C_k^{(r)})$ be the maximum numb…

For every integer $t\ge 1$ and every graph $G$ with $\chi(G)\ge t$, there exist pairwise vertex-disjoint connected subgraphs $T_1,\dots,T_t$ such that, for every pair…

Let $\mathcal{I}(n,k)$ denote the set of intersecting families $\mathcal{F}\subseteq\binom{[n]}{k}$, and let a family be trivial if all its members contain a common element. Then,…

For every fixed $\varepsilon>0$, determine whether the random graph $G(n,p)$ has, with high probability, a fractional triangle decomposition precisely at the conjectured threshold…

Let $\mathcal{P}(n,k)$ be the set of integer partitions of $n$ into exactly $k$ positive parts. For $1\le t<k$, call a family $\mathcal{A}\subseteq\mathcal{P}(n,k)$ $t$-intersectin…

Let $F$ be an ordered forest on $n$ vertices, and let $\chi_{<}(F)$ denote its interval chromatic number. Is there a constant $C>0$ such that every such $F$ satisfies … Here…

For which integers $r\ge 0$ does there exist a finite simple $r$-regular graph $G$ such that, for every vertex $v\in V(G)$, the triangle-degree…

For integers $r,k\ge 2$ and $s\ge r$, let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $r$-uniform hypergraph $H$ on $n$ vertices such that no set of $k$ distinct edges ha…

For every integer $k\ge 2$, if $\mathcal A\subseteq 2^{[2k]}$ is an intersecting temperate family, then … Moreover, equality is conjectured to hold only for a family of the followi…

For the tetrahedron $K_4^{(3)}$, determine whether its Turán density satisfies $\pi(K_4^{(3)})=\frac{5}{9}$. Equivalently, if $\operatorname{ex}(n,K_4^{(3)})$ denotes the maximum n…

For every pair of integers $s>k\ge 4$, there exists a constant $c=c(s,k)>0$ such that, for all sufficiently large integers $n$, $r_k(s,n)\ge \operatorname{twr}_{k-1}(cn)$. Here…

For every integer $n\ge 6$, let $m=\binom{n}{2}$ and consider the incidence family of the complete graph $K_n$, with one column for each vertex and one row for each edge. If…

For positive integers $n$ and $t$ with $n\ge 5t$, every tripartite graph $G$ whose three parts each have size $n$ and whose minimum degree satisfies $\delta(G)\ge n+t$ has at least…

For every acyclic matrix pattern $P$, there exists a constant $C_P$ such that the extremal function satisfies $\operatorname{Ex}(P,n)=O\!\left(n(\log n)^{C_P}\right)$ as…