58 problems

Mubayi–Rödl conjecture.

Peng–Zhao's conjecture. The Lagrangian of $G$ satisfies

Erdős matching conjecture. The number of edges of $H$ satisfies

Let $3\leq k<n$, and let $K_n^{(k)}$ be the complete $k$-uniform hypergraph on $n$ vertices, with $\binom nk$ edges. A Hamilton Berge cycle is a Berge cycle of length $n$ using all…

Let $k$ be an odd integer, and let $H$ be an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs. A full $k$-star consists of all $k$-edges containing a fixed cen…

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 $\widetilde{m}(n,k,k,2,2)$ be the maximum product of the sizes of two cross-intersecting $k$-uniform families, each having covering number $2$. The extension conjecture. The fo…

Let $m(n,k,s)$ denote the maximum size of a $k$-uniform hypergraph on an $n$-element vertex set whose covering number is $s$. For $s\geq 3$, $k>k_0(s)$, and $n\geq n_0(k,s)$, the a…

Let $\mathcal{C}_\ell^r$ be the tight $r$-uniform cycle of length $\ell>r$, with vertex set $\{0,1,\ldots,\ell-1\}$ and edges … where addition is modulo $\ell$. For $r=3$,…

Let $\mathcal{H}$ be an $r$-uniform hypergraph, and for a family $\mathcal{F}$ of $r$-graphs let $\mathrm{ex}_{r-1}(n,\mathcal{F})$ be the maximum possible minimum codegree…

Let $r$ be the uniformity, let $H$ be a sufficiently dense $r$-uniform hypergraph, and let $\rho(H)$ denote the density parameter used in the paper. Sharp threshold conjecture. The…

Let $r$ be the uniformity, let $H$ be a sufficiently dense $r$-uniform hypergraph, and let $\rho(H)$ denote the parameter used to describe its density. Sharp threshold conjecture.…

For integers $r\geqslant 3$ and $\ell\geqslant 4$, let $\operatorname{Forb}_r^L(n,C_\ell^r)$ denote the family of $n$-vertex linear $r$-uniform hypergraphs containing no linear cyc…

The iterated-partite characterization. $H$ is iterated $k$-partite if and only if

Tight tree Ramsey growth conjecture. For $r\geq 5$, if $T$ is a non-trivial tight $r$-tree, then there exist constants $c_1,c_2>0$ such that, for every positive integer $n$,

The family-level tower lower-bound conjecture. If $k>s$, then there exists a positive constant $c$ such that

The tower lower-bound conjecture. If $k>s$ and $H$ is an $s$-tightly connected $k$-graph that is not $k$-partite, then there exists a positive constant $c$ such that

The iterated-tripartite classification conjecture. For a 3-graph $H$, there exists a constant $c$ depending only on $H$ such that

Let $H$ be a connected $k$-uniform hypergraph, let $\mathcal{L}(H)$ be its Laplacian tensor, let $\mathcal{A}(H)$ be its adjacency tensor, and let $rho$ be the spectral radius of…

Let $H$ be a connected $k$-uniform hypergraph, let $\mathcal{A}(H)$ be its adjacency tensor, let $rho$ be its spectral radius, and let $\mathbb{V}_{rho}(\mathcal{A}(H))$ be the pro…

Let $r\geq 3$, and let $M_k$ be a matching consisting of $k$ pairwise disjoint $r$-edges. Write ${\rm ar}(n,r,M_k)$ for the anti-Ramsey number and ${\rm ex}(n,r,M_{k-1})$ for the T…

Özkahya and Young's conjecture.

Graft transformation conjecture. For $s\geq t\geq 2$,

Zheng's conjecture. For every $k$-uniform hypertree $T$ with $k\geq 3$, the multiplicity of the zero Laplacian eigenvalue is $k^{|E(T)|(k-2)}$. This conjecture concerns the Laplaci…

Convergence-equivalence conjecture. Left-convergence, partitionable convergence, and action convergence of the $(r-1)$-action of the sequence