213 problems

For integers $r\ge 3$ and $r-2\ge t\ge 1$, let $\iota_0(t,n;r)$ denote the maximum size of a $t$-intersecting family in an $r$-partite $r$-uniform hypergraph with all parts of size…

Polynomial bound conjecture. We have

Rank-$n+1$ hypergraph conjecture. If an $n$-party inequality contracts on $(n+1)$-graphs, then it is a valid inequality on all hypergraphs of finite rank.

The truncate-and-relax threshold conjecture. If $I_2(\alpha,\beta) > 1$, then

Let $\mathcal{HA}_n$ be the poset of hyperarbres, and let $\operatorname{WH}_i(\mathcal{HA}_n)$ denote its Whitney homology groups. Define the symmetric-function generating series…

Frankl's conjecture. There exists a constant $c$ such that, whenever $k>c d^2$,

Let $f(n,k)$ denote the maximal-clique deficiency parameter and let $c(n,k)$ denote the layered-tree parameter for $k$-uniform hypergraphs, as defined in the paper. Exponential rel…

The $P_4^r$ upper-bound conjecture.

Let $G=(\mathcal{V},\mathcal{E})$ be a simplicial complex. A triangle edge choice function assigns to each triangle of $G$ one of its edges, and it is injective when distinct trian…

Let $k\geq 3$ be an integer. A $k$-uniform hypergraph is linear if it contains no $2$-cycle, and let $\alpha({\mathcal H})$ denote its independence number. Shattering-threshold con…

Let $k\geq 3$, and let $a_k>0$ satisfy $a_k=1-o_k(1)$ as $k\to\infty$. A $k$-uniform hypergraph is linear if it contains no $2$-cycle. Verstraëte–Wilson conjecture. Every $n$-verte…

A pseudorandom class of hypergraphs is a hereditary class whose sparse random model contains, with high probability, an induced subhypergraph on at least $(1-o_d(1))n$ vertices bel…

Let $m(k,k)$ denote the maximum product of the sizes of two cross-intersecting families with maximal covering number $k$ in the setting of the paper. The diagonal conjecture. For a…

Let $G$ be a $k$-chromatic $r$-graph of order $n>(r-1)k$, where $k\geq 2$ and $r\geq 4$. For $p\geq 1$, let $\lambda^{(p)}(G)$ denote its $p$-spectral radius. Kang–Nikiforov–Yuan's…

Let $G$ be an $r$-uniform hypergraph, and let $\binom{n}{r}$ denote the number of $r$-subsets of an $n$-vertex set. For $p\geq 1$, let $\lambda^{(p)}(G)$ be the maximum of the poly…

Jung–Keszegh–Pálvölgyi–Yuditsky conjecture. For all $k\geq2$ there exists a constant $c_k<1$ such that, whenever

Core-structure conjecture. Under these hypotheses, every vertex incident with more than linearly many hyperedges shares a second common vertex with all those hyperedges.

Let $H$ be an $r$-uniform $d$-degenerate triangle-free hypergraph, with $r\geq 3$. Here, $\chi_f(H)$ denotes the fractional chromatic number, and $c_r>0$ is a constant depending on…

Let $q,s\ge 1$ and $r\ge 2$ be integers. An $s$-connector is an $r$-uniform hypergraph such that every collection of pairwise disjoint vertex sets $A_1,\dots,A_r$ with $|A_i|\ge s$…

Let $\Gamma$ be a hypergraph with non-negative weights. For every $k\in\mathbb{Z}_{\geq 1}$, let $\omega_k=\omega_k(\Gamma)$ denote the smallest new eigenvalue of the Laplacian…

Let $\Gamma=([n],w)$ be an arbitrary hypergraph with non-negative weights. For each irreducible representation $\rho$ of $\mathrm{U}(n)$, let $\lambda_{\min}(\Gamma,\rho)$ denote t…

Let $\Gamma=([n],w)$ be a weighted hypergraph on $[n]$, with non-negative weights, and let $\rho$ be a non-trivial irreducible representation of $\mathrm{Sym}(n)$. Write…

Illingworth–Lang–Müyesser–Parczyk–Sgueglia's conjecture. If $H$ has minimum codegree at least $n/k$, then $H$ has a spanning tight component.

Győri et al.'s linear Turán conjecture. If $G$ contains no linear path of length $k\ge 5$, then the number of edges in $G$ is at most

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