90 problems

Erdős matching conjecture. The extremal function satisfies

Erdős Matching Conjecture. Let $n\geq (k+1)r$. If $H$ is an $M_{k+1}$-free $r$-graph on $n$ vertices, then

Let $H$ be a $k$-uniform hypergraph on $n$ vertices. For $0\leq \ell\leq k-1$, let $m_\ell(k,n)$ be the smallest integer $m$ such that every $k$-graph on $n$ vertices with minimum…

Exact spectral threshold conjecture. If

A Steiner triple system of order $n$ is a $3$-uniform hypergraph on $n$ vertices in which every pair of vertices lies in exactly one edge. A matching is a set of pairwise vertex-di…

Let $H$ be a $k$-uniform hypergraph, let $M_s$ denote a matching with $s$ edges, let ${\rm ar}(n,k,M_s)$ be the anti-Ramsey number of $M_s$ in the complete $k$-uniform hypergraph o…

Let $H$ be a 3-graph of order $n$, let $sigma_2(H)$ be the minimum of $deg(u)+deg(v)$ over adjacent vertices $u,v$, and let $H_{n,s}^2$ be the corresponding extremal 3-graph. Zhang…

Let $k\geq 2$ and let $H=(V,E)$ be a $k$-uniform hypergraph on $n$ vertices. For a $(k-1)$-subset $S\subseteq V$, write $\deg_H(S)$ for the number of edges containing $S$, and let…

Let $H$ be a $k$-graph, let $d$ be an integer with $1\leq d\leq k-1$, and let $m_d^s(k,n)$ denote the minimum integer $m$ such that every $n$-vertex $k$-graph $H$ with minimum $d$-…

Wanless's conjecture. Every partite $(d,d+1,n)$-Steiner system has a perfect matching if $n$ or $d$ is even.

Let $\mathcal{F}$ be a $k$-graph on vertex set $[n]$. Let $\nu(\mathcal{F})$ denote its matching number and let $\tau(\mathcal{F})$ denote the minimum size of a vertex cover, meani…

Erdős matching conjecture. The stated edge bound should hold. The conjecture gives the extremal edge condition forcing a matching of a prescribed size. Its case $s=0$ was resolved…

Let $k$ be a positive integer, let $H$ be a $k$-graph with $n=|V(H)|$ vertices, and suppose that $n\not\equiv 0\pmod{k}$. A matching $M$ in $H$ is near-perfect if…

Let $G$ be a $k$-uniform hypergraph on $n$ vertices, and let $m_0^s(k,n)$ be the minimum number of edges such that every such hypergraph has a matching of size $s$. Erdős's matchin…

Let $k,\ell\in\mathbb{N}$ with $1\leq \ell<k$, and define $c_{k,\ell}^{*}$ to be the smallest number $c$ such that every $k$-uniform hypergraph $H$ on $n$ vertices with … contains…

Let $\mathcal{F}$ be a $k$-uniform hypergraph on $n$ vertices, and let $\sigma_k(\mathcal{F})$ denote its Ore-degree, the minimum of $\deg(S)=\sum_{v\in S}\deg(v)$ over all non-edg…

Local spectral matching conjecture. If, for every $v\in V(H)$,

For integers $r\ge 3$ and $n_1\ge\dots\ge n_r\ge 2$, let $X_1,\dots,X_r$ be disjoint sets with $|X_i|=n_i$. Define $m_0(s,n_1,\dots,n_r)$ as the maximum size of a family…

Ore-degree Erdős Matching Conjecture. If $n>rs$ and

Let $n,s,k$ be positive integers, and let $\mathcal F\subseteq[n]^k$. Two tuples in $[n]^k$ are disjoint if they differ in every coordinate. Partite Erdős matching conjecture. If ……

Let $n,m,k$ be positive integers with $n$ sufficiently large, $n>m$, and $k\geq 4$. Let $H_k(n,m)$ be the $k$-partite $k$-graph constructed by taking pairwise disjoint partition cl…

Let $H$ be a $k$-uniform hypergraph, and for $1\leq \ell<k$ let $\delta_\ell(H)$ denote its minimum $\ell$-degree. Define $m_{\ell,k}$ as the infimum of those $c\in[0,1]$ such that…

Matching existence conjecture. If this conjecture holds for $6\leq k\leq 16$, then

Let a Steiner triple system of order $n$ be a $3$-uniform hypergraph on $n$ vertices in which every pair lies in exactly one edge, and let a perfect matching be a set of $n/3$ pair…

Let $H$ be a $k$-uniform hypergraph. For $1 \leq m \leq k-1$, let $\nu^{(m)}(H)$ be the maximum size of a set of edges of $H$ whose pairwise intersections have size less than $m$,…