11 problems

Let $\mathcal{T}$ be a tight $k$-tree on $v$ vertices, meaning a hypertree of $k$-sets in which consecutive edges intersect in $k-1$ elements, and $|\bigcup\mathcal{T}|=v$. Let…

Let $H$ be an $r$-uniform hypergraph on $n$ vertices containing no $(k+1)$-connected subgraph. Hypergraph Mader conjecture. For all sufficiently large $n$, … This proposed extensio…

Let $q>r\geq 3$ be integers. An $r$-graph is an $r$-uniform hypergraph, and it is $K_q^r$-divisible when it satisfies the divisibility conditions necessary for a decomposition into…

Let $\mathcal{F}\subseteq\binom{[n]}{k}$ be $3$-cluster-free, meaning it contains no three distinct $k$-sets with empty common intersection and union of size at most $2k$. A comple…

Higher-uniformity Zarankiewicz conjecture. For $k>2$, the exponents of $u$ and $D$ in the semialgebraic Zarankiewicz bound can be decreased so that the bound is tight in the sense…

Localized hypergraph clique-weight conjecture. One has

Let an $r$-graph be an $r$-uniform hypergraph on $n$ vertices, and let a tight cycle of length $\ell$ be the hypergraph formed from $\ell$ cyclically ordered vertices by taking the…

A $3$-uniform hypergraph is a hypergraph whose edges have size $3$. A $2$-core is a hypergraph with minimum degree at least $2$. Solymosi–Solymosi conjecture. Every sufficiently la…

Let $r\geq 2$. An injective $q$-coloring of an $r$-uniform hypergraph is a coloring of its vertices with $q$ colors such that any two edges sharing $r-1$ vertices have distinct col…

Let $r\geq 2$ and let $2\leq s_1\leq s_2\leq\cdots\leq s_r$. Write $K_{s_1,\ldots,s_r}$ for the complete $r$-partite $r$-uniform hypergraph with parts of these sizes, and let…

Let $[n]=\{1,\ldots,n\}$ and let $M(\ell,n,k)$ be the maximum cardinality of a family ${\cal A}\subset {[n]\choose k}$ containing no $\ell$ pairwise disjoint members. Equivalently,…