46 problems

Exact spectral threshold conjecture. If

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 $H$ 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 $d$-…

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…

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 $k\geq 3$ be a positive integer and let $0<\delta,\varepsilon<1$ be positive real numbers. There is an $n_0$ such that, for every $n\geq n_0$, if $H$ is an $n$-vertex $k$-graph…

Let $k\geq 2$ be an integer, let $H$ be an $n$-vertex $d$-regular linear $k$-graph, and let $M$ be a matching chosen uniformly from the set of all matchings of $H$. For a vertex…

Let $H$ be a $d$-regular linear $k$-graph, and let $M$ be a matching chosen uniformly from the set of all matchings of $H$. For a vertex $v$, write $\mathcal{P}_H(\overline{v})$ fo…

Let $H$ be a 3-graph of order $n$, let $\sigma_2(H)$ denote the minimum $\deg(u)+\deg(v)$ over adjacent vertices $u,v$, and let $H^{1,2}_{n,x,y}$ be the construction defined in the so…

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…

For a hypergraph $H$, a 2-matching is a set of edges such that every two edges share fewer than two vertices, and let $\nu^{(2)}(H)$ denote its maximum size. For a Young diagram…

Frankl–Kupavskii stability conjecture. If $\nu(F)\le t$, then $\tau(F)\le t$ or

Let $n,s,k$ be positive integers with $k\geq 2$ and $n\geq ks+k-1$. Let $H$ be a $k$-graph on vertex set $[n]$, and let $\nu(H)$ and $\tau(H)$ denote its matching number and vertex…

Good-function conjecture. If

Huang–Zhao conjecture. For positive integers $m,k,n$ satisfying $m<n/k$, if

Fractional Erdős matching conjecture. For integers $\ell,d\geq1$,

Aharoni–Howard's conjecture. If each $H_i$ has $f(n,r,k)$ hyperedges, then there is a rainbow matching

Let $\mathcal F\subset {[n]\choose k}$ satisfy $\nu(\mathcal F)\leq s$ and $n\geq k(s+1)$. Define the $s$-diversity by … where…

Fix an integer $k\geq2$ and a positive real number $x\leq1/k$. Let $m(n,k,xn)$ denote the maximum size of a $k$-uniform family on $[n]$ whose matching number is less than $xn+1$. A…

Let $H$ be a $3$-graph of order $n$, let $s$ be an integer with $2\le s\le n/3$, and let $H_{n,3,s}^2$ and $H_{n,3,s}^3$ be the two extremal $3$-graphs used in the degree-sum compa…

$K_4^3$-free matching conjecture. For any $\gamma>0$, for sufficiently large $n\in 3\mathbb{N}$, if $H$ contains no copy of $K_4^3$ and

Let $\mathcal{H}$ be a simple $3$-partite $3$-graph with vertex classes $A$, $B$ and $C$. Suppose each vertex in $A$ has degree at least $r$, and each vertex in $B \cup C$ has degr…

Let $\mathcal{H}$ be an $r$-regular simple $3$-partite $3$-graph with $n$ vertices in each class. Aharoni–Kotlar–Ziv conjecture. … This conjecture concerns the matching number of s…