Intersecting families of loose spanning hypertrees
Abstract
A loose hypertree is a hypergraph whose incidence graph is a tree. Let b≥3, n≡1pmodb−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≤ t≤ 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≤ n, the extremal forests are matchings. We also prove a Hilton--Milner-type bound: if n≥ Cb(t+1), where Cb depends only on b, every nontrivial t-intersecting family has at most (s+1)rt+1(n) members. Here rj(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 Ob,t(n2−b)rt(n). The proofs combine exact forest enumeration with the intersection theorems of Kupavskii for spread families.
Related
BibTeX
@misc{person2026intersecting,
title = {{Intersecting families of loose spanning hypertrees}},
author = {Yury Person and Thomas Schweser},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20443},
eprint = {2610.20443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20443},
}
RIS
TY - UNPB
TI - Intersecting families of loose spanning hypertrees
AU - Yury Person
AU - Thomas Schweser
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20443
UR - https://arxiv.org/abs/2610.20443
AB - A loose hypertree is a hypergraph whose incidence graph is a tree. Let b≥3, n≡1pmodb−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≤ t≤ 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≤ n, the extremal forests are matchings. We also prove a Hilton--Milner-type bound: if n≥ 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.
ER -