Matching and intersection problems for non-trivial $r$-partite $r$-uniform hypergraphs

TL;DR

This paper determines exact bounds for non-trivial r-partite r-uniform hypergraphs concerning matching and intersection problems, confirming parts of a conjecture for large n and specific t values.

Contribution

It provides the first exact bounds for non-trivial r-partite hypergraphs in these problems and resolves specific cases of the intersection problem.

Findings

Exact bounds for matching and intersection problems when n is large

Resolution of the t=1 and t=r-2 cases for all n ≥ 2

Partial confirmation of Lu and Ma's conjecture

Abstract

A central theme in extremal combinatorics is the study of the maximum number of edges in an $r$-uniform hypergraph ($r$-graph) with matching number at most $s$ (the Erd\H{o}s Matching Conjecture) or with pairwise intersection at least $t$ (the $t$-intersection problem). The maximum sizes for these problems are typically achieved by trivial constructions: for the matching problem, the extremal construction consists of all edges intersecting a fixed set of $s$ vertices, while for the intersection problem, it consists of all edges containing a fixed set of $t$ vertices. In this paper, we investigate the \emph{non-trivial} $r$-partite $r$-graphs where each part is of size $n$. We determine the exact bounds for both the matching problem and the intersection problem when $n$ is sufficiently large. Furthermore, for the intersection problem, we resolve the cases $t=1$ and $t=r-2$ for all $n \ge 2$. Our results partially confirm a conjecture of Lu and Ma.