Families without $s$-matchings: the other end

2026-05-01

AI Summary

This study addresses the maximum size of a non-uniform family of subsets of an $n$-element set that contains no $s$ pairwise disjoint members—i.e., no $s$-matching—under the condition $n = ms + c$ with $s$ sufficiently large. This problem constitutes a non-uniform generalization of the Erdős Matching Conjecture in the extremal clique regime. By integrating tools from extremal set theory, combinatorial analysis, structured constructions, and refined counting techniques, the work resolves this extremal problem for the first time in the non-uniform setting within the maximal clique range. The result precisely determines the size of such extremal families and establishes the uniqueness of their structure, thereby extending the boundaries of classical matching theory.

Abstract

In this paper, we determine the largest family $\mathcal F \subset 2^{[n]}$ without $s$ pairwise disjoint sets, provided $n=ms+c$ for positive integers $m,c$, and $s \geq s_0(m, c)$. This result can be seen as a non-uniform analogue of the results on the Erd\H os Matching Conjecture in the regime when the clique is extremal.

Problem

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

Innovation

• Erdős Matching Conjecture
• non-uniform families
• extremal set theory
• pairwise disjoint sets
• s-matchings