A generalization of Erd\H{o}s' matching conjecture

TL;DR

This paper generalizes Erd ext{"o}s' matching conjecture by determining the maximum edges in hypergraphs with a bounded $k$-matching number, proposing candidate extremal hypergraphs and establishing conditions for optimality.

Contribution

It extends the classical matching conjecture to a broader $k$-matching context and identifies candidate extremal hypergraphs under specific size conditions.

Findings

Proposes candidate extremal hypergraphs for the generalized problem.

Shows extremal hypergraph is among candidates when $n \\ge 4r\\binom{r}{k}^2\\cdot a$.

Generalizes the $k$-intersection problem with new bounds.

Abstract

Let $\mathcal{H}=(V,\mathcal{E})$ be an $r$-uniform hypergraph on $n$ vertices and fix a positive integer $k$ such that $1\le k\le r$. A $k$-\emph{matching} of $\mathcal{H}$ is a collection of edges $\mathcal{M}\subset \mathcal{E}$ such that every subset of $V$ whose cardinality equals $k$ is contained in at most one element of $\mathcal{M}$. The $k$-matching number of $\mathcal{H}$ is the maximum cardinality of a $k$-matching. A well-known problem, posed by Erd\H{o}s, asks for the maximum number of edges in an $r$-uniform hypergraph under constraints on its $1$-matching number. In this article we investigate the more general problem of determining the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices subject to the constraint that its $k$-matching number is strictly less than $a$. The problem can also be seen as a generalization of the, well-known, $k$-intersection problem. We propose candidate hypergraphs for the solution of this problem, and show that the extremal hypergraph is among this candidate set when $n\ge 4r\binom{r}{k}^2\cdot a$.