More on the Erd\H os--Kleitman problem on matchings in set families

2026-05-05

Abstract

Let $e(n,s)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. In 1968, answering a question of Erdős, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integers $m,s\ge 1$. Half a century later, Frankl and Kupavskii determined $e(s(m+1)-\ell, s)$ for $\ell \leq \frac{s-3}{m+3}$. They showed that the corresponding extremal example is closely connected with the extremal example for the Erdős Matching Conjecture, and conjectured that the same remains true for all $\ell \leq s/2$. In this paper, we prove an approximate version of their conjecture for $s\ge s_0(m)$.

Problem

• Erdős–Kleitman problem
• matchings
• set families
• extremal combinatorics
• disjoint subsets

Innovation

• Erdős Matching Conjecture
• set families
• pairwise disjoint sets
• extremal combinatorics
• approximate result