A complete solution of the Erdős-Kleitman matching problem for $n\le 3s$

TL;DR

The paper resolves the Erdős–Kleitman non-uniform matching problem e(n,s) for all n ≤ 3s by identifying four extremal shifted up-set families and proving that the maximum size equals the largest among their complements, e(n,s)=max{|al P(s,ℓ)|,|al P'(s,ℓ)|,|al Q(s,ℓ)|,|al W(s,ℓ)|} for n=2s+c=3s−ℓ with suitable parameter ranges. The authors develop a unified framework around the d(F) parameter and bounds on the 2- and 3-element layers y(2), y(3), using an averaging argument over carefully chosen matchings to constrain the structure of extremal families. A detailed case analysis handles large, moderate, and small d, with c ≥ 5 treated in the main text and c ≤ 4 in an appendix, culminating in a complete classification of extremals in the specified regime. This work advances understanding of the EMC landscape, provides exact values in a broad new range, and demonstrates the pivotal role of four structured extremal families in the non-uniform setting.

Abstract

Given integers $n\ge s\ge 2$, 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 $m,s\ge 1$. The question of determining $e(n,s)$ is closely connected to its uniform counterpart, the subject of the famous Erdős 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 $n\le 3s$. 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.

Figures (2)

• Figure 1: Extremal families. $\mathcal{W}$ is one of the extremal families for $c = 1, s \geq 5$.
• Figure 2: Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\ell = 1$, that is, $s = c + 1$, the family $\mathcal{P}'$ coincides with $\mathcal{P}$ (both are equal to ${n \choose \geq (m + 1)}$) and thus is formally extremal. When $\mathcal{P}' \neq \mathcal{P}$, for $c \leq 9$$\mathcal{P}'$ cannot be extremal.

Theorems & Definitions (71)

• Conjecture 1: Erdős Matching Conjecture E
• Theorem 1: Kleitman Kl
• Definition 1
• Conjecture 2: FK9
• Theorem 2: FK9FK8
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Corollary 6