Families with no perfect matchings

TL;DR

This paper investigates the maximum size of families of k-subsets of {1,...,n} with no perfect matching and no small blocking set, proving optimality of a known construction for large k and n.

Contribution

It proves the optimality of Frankl's construction for large k (≥100) and sufficiently large n, extending previous results and resolving a conjecture.

Findings

Frankl's construction is optimal for k ≥ 100 and large n.

The paper confirms the conjecture for large k and n.

It extends the understanding of matchings in combinatorial families.

Abstract

We consider families of $k$-subsets of $\{1, \dots, n\}$, where $n$ is a multiple of $k$, which have no perfect matching. An equivalent condition for a family $\mathcal{F}$ to have no perfect matching is for there to be a blocking set, which is a set of $b$ elements of $\{1, \dots, n\}$ that cannot be covered by $b$ disjoint sets in $\mathcal{F}$. We are specifically interested in the largest possible size of a family $\mathcal{F}$ with no perfect matching and no blocking set of size less than $b$. Frankl resolved the case of families with no singleton blocking set (in other words, the $b=2$ case) for sufficiently large $n$ and conjectured an optimal construction for general $b$. Though Frankl's construction fails to be optimal for $k = 2, 3$, we show that the construction is optimal whenever $k \ge 100$ and $n$ is sufficiently large.