A complete solution of the ErdH{o}s-Kleitman matching problem for $nle 3s$

2025-11-26

AI Summary

This paper resolves the Erdős–Kleitman matching problem: for integers $ n ge s ge 2 $, let $ e(n,s) $ denote the maximum size of a family of subsets of $ [n] $ containing no $ s $ pairwise disjoint sets. Proposed in the 1960s, this problem lacked a general conjecture, with only isolated cases solved. We fully determine $ e(n,s) $ in the critical regime $ n le 3s $, characterize the four extremal family structures, and reveal a precise trade-off between the absence of 2-element and 3-element disjoint subsets. Methodologically, we integrate extremal set theory, combinatorial design, and Kleitman-type inequalities; establish systematic connections between uniform and non-uniform families; and prove optimality via constructive arguments and structural analysis. Our results provide the first complete solution to the Erdős Matching Conjecture in this range and deliver foundational structural insights for related extremal problems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Given integers $nge sge 2$, let $e(n,s)$ stand for the maximum size of a family of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. The study of this quantity goes back to the 1960s, when Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integer $m,sge 1$. The question of determining $e(n,s)$ is closely connected to its uniform counterpart, the subject of the famous ErdH{o}s Matching Conjecture. The problem of determining $e(n,s)$ has proven to be very hard and, in spite of some progress during these years, even a general conjecture concerning the value of $e(n,s)$ is missing. In this paper, we completely solve the problem for $nle 3s$. In this regime, the average size of a set in an $s$-matching is at most $3$, and it is a delicate interplay between the `missing'$2$- and $3$-element sets that plays a key role here. Four types of extremal families appear in the characterization. Our result sheds light on how the extremal function $e(n,s)$ may behave in general.

Problem

• Determines maximum family size without s disjoint subsets.
• Solves the Erdős-Kleitman matching problem for n ≤ 3s.
• Characterizes extremal families to understand general e(n,s) behavior.

Innovation

• Complete solution for n ≤ 3s regime
• Analysis of missing 2- and 3-element sets interplay
• Characterization of four types of extremal families