Intersecting families of loose spanning hypertrees

Abstract

A loose hypertree is a hypergraph whose incidence graph is a tree. Let $b\ge 3$, $n\equiv 1(\mathrm{mod}b-1)$, and $s:=(n-1)/(b-1)$. We determine the largest $t$-intersecting families of loose spanning $b$-uniform hypertrees on $[n]$, simultaneously for every $1\le t\le s$, provided $n$ exceeds an explicit constant depending only on $b$. The extremal families consist of all such hypertrees containing a fixed $t$-edge loose forest whose component sizes are as equal as the divisibility condition permits. In particular, when $bt\le n$, the extremal forests are matchings. We also prove a Hilton--Milner-type bound: if $n\ge C_b(t+1)$, where $C_b$ depends only on $b$, every nontrivial $t$-intersecting family has at most $(s+1)r_{t+1}(n)$ members. Here $r_j(n)$ is the maximum possible number of loose spanning $b$-uniform hypertrees containing a fixed $j$-edge loose forest. For fixed constants $b,t$, this bound is asymptotically sharp and is $O_{b,t}(n^{2-b})r_t(n)$. The proofs combine exact forest enumeration with the intersection theorems of Kupavskii for spread families.

arXiv comment12 pages