300 problems

For a $3$-uniform hypergraph $F$, let $\operatorname{ex}_3(n,F)$ be the maximum number of edges in an $F$-free $3$-graph on $n$ vertices, and define its Turán density by … Let…

Ryser's conjecture. The size of a minimum vertex cover of $H$ is at most $r-1$ times the size of a maximum matching of $H$.

Let $K_4^{(3)}$ be the complete $3$-uniform hypergraph on $4$ vertices, and let $K_4^-$ be its $3$-edge subgraph. Write $\pi(K_4^-)$ for the Turán density of $K_4^-$. Frankl–Füredi…

Pósa-type conjecture for triple systems. For every $\varepsilon>0$, there is $n_0$ such that every $3$-uniform hypergraph $G$ of order $n\geq n_0$ satisfying

Let $\mathcal{P}^k_n$ and $\mathcal{C}^k_n$ denote the $k$-uniform loose path and loose cycle with $n$ edges, respectively, and let $R(H_1,H_2)$ be the two-colour Ramsey number for…

Let $r\ge 2$. An $r$-graph is a hypergraph whose edges have size $r$. Let $\mathbb{C}_{r}$ be the collection of $r$-graphs consisting of three edges $A,B,C$ such that … Let…

Elliott–Rödl conjecture. Every hypertree of $(1-o(1))n$ vertices can be found in any Steiner triple system.

For $k,\ell\in\mathbb N$ with $\ell\leq k-1$, let $c^*_{k,\ell}$ be the smallest number $c$ such that every $k$-graph $H$ on $n$ vertices with … contains a perfect fractional match…

Let $H$ be an $r$-partite hypergraph containing at least one edge, and let $\nu(H)$ denote its matching number, the maximum number of pairwise disjoint edges. For a set $S$ of vert…

Let $H$ be an $n$-vertex $k$-uniform hypergraph, and let $\delta_{k-1}(H)$ denote its minimum codegree. A tight Hamilton cycle is a Hamilton $\ell$-cycle with $\ell=k-1$, equivalen…

Large-degree conjecture. There exists $d_r>0$ such that if $H$ has an edge of multiplicity at least $\nu+1$, then

Let $\mathbb{S}$ be the parameter matrix of a $k$-transversal in a $d$-uniform $r$-regular hypergraph $\mathcal{G}$. Let $\xi$ be a $d$-th primitive root of unity. Characteristic-p…

Mubayi's conjecture. If $\mathcal F$ is a stable non-$r$-partite $r$-graph, then, for every positive integer $q$ and all sufficiently large $n$, every $n$-vertex $r$-graph with…

Han–Zhao's conjecture. If

Let $K_{s,t}^{(3)}$ denote the complete $(s,t)$-partite $3$-uniform hypergraph, and let $\mathrm{ex}(n,K_{s,t}^{(3)})$ be the maximum number of edges in an $n$-vertex…

Let $\mathcal H\subseteq\binom{[n]}{k}$ be a $k$-uniform hypergraph, and let $\mathcal C\subseteq\mathcal H$ be an even cover if $\mathcal C$ is nonempty and every vertex lies in a…

Hypergraph Nash–Williams–Tutte conjecture. For positive integers $t$ and $k$, every $k$-weakly-partition-connected hypergraph $H$ on $t$ vertices has a $k$-distinguishable tree ass…

Let $k$ be the ground field, and let $\Lambda_{V_d}^{S^r}$ be the exterior-algebra-like construction associated with the $d$-dimensional vector space $V_d$. Write…

Iterated-logarithm conjecture. The number $cell-1$ of iterated logarithms is best possible:

Erdős's conjecture.

Let $C_{13}$ denote the crown on 13 vertices, and let $ex^{lin}_3(n,C_{13})$ be the maximum number of edges in a linear 3-uniform hypergraph on $n$ vertices that contains no copy o…

Keevash–Knox–Mycroft conjecture. For $1\leq\ell<k$, $\textbf{PM}(k,\ell,\delta)$ is in P for every

Aharoni–Zerbib's equality conjecture. These functions should satisfy

Erdős–Lovász conjecture. The size of the intersection spectrum tends to infinity with $k$:

Sharp second-order conjecture. Suppose $r,d\geq 2$. Then