[Paper Review] Two problems of P. Erdős on matchings in set families.

Péter Frankl, Andrey Kupavskii|arXiv (Cornell University)|Jul 20, 2016

Limits and Structures in Graph Theory3 citations

TL;DR

This paper resolves a long-standing problem in extremal combinatorics by determining the maximum sum of sizes of $ s $ set families on $[n]$ that are $ q $-dependent, generalizing Kleitman's theorem. It establishes a Hilton-Milner-type stability result for the Erd\'os Matching Conjecture in a wide range, enabling the exact determination of the anti-Ramsey number $ ar(n,k,s) $ for $ n \geq sk + (s-1)(k-1) $ and $ k \geq 3 $.

ABSTRACT

The families $\mathcal F_1,\ldots, \mathcal F_s\subset 2^{[n]}$ are called extit{$q$-dependent} if there are no pairwise disjoint $F_i\in \mathcal F_i, i=1,\ldots, s,$ satisfying $|F_1\cup\ldots\cup F_s|\le q.$ We determine $\max |\mathcal F_1|+\ldots +|\mathcal F_s| $ for extit{all} values $n\ge q,s\ge 2$. The result provides a far-reaching generalization of an important classical result of Kleitman. The uniform case $\mathcal F_1 = \ldots = \mathcal F_s\subset {[n]\choose k}$ of this problem is the so-called Erd\H os Matching Conjecture. After more than 50 years its full solution is still not in sight. In the present paper we provide a Hilton-Milner-type stability theorem for it in a relatively wide range, in particular, for $n\ge (2+o(1))sk$ with $o(1)$ depending on $s$. This is a considerable improvement of a result due to Bollob\'as, Daykin and Erd\H os. We apply our results to advance in an anti-Ramsey-type problem, proposed by \Ozkahya and Young. They asked for the minimum number $ar(n,k,s)$ of colors in the coloring of the $k$-element subsets of $[n]$ that do not contain a extit{rainbow matching} of size $s$, that is, $s$ sets of different colors that are pairwise disjoint. We prove a stability result for the problem, which allows to determine $ar(n,k,s)$ for all $k\ge 3$ and $n\ge sk+(s-1)(k-1).$ Some other consequences of our results are presented as well.

Motivation & Objective

• To determine the maximum sum of sizes of $ s $ set families on $[n]$ that are $ q $-dependent for all $ n \geq q, s \geq 2 $.
• To provide a Hilton-Milner-type stability result for the uniform Erd\'os Matching Conjecture in the range $ n \geq (2+o(1))sk $, improving prior bounds.
• To resolve an anti-Ramsey-type problem by determining the minimum number of colors $ ar(n,k,s) $ needed to avoid a rainbow matching of size $ s $ in $ k $-sets.
• To extend the applicability of stability methods to extremal set systems with structural constraints.

Proposed method

• Introduce the concept of $ q $-dependence to characterize families with no $ s $ pairwise disjoint sets whose union has size at most $ q $.
• Use a novel stability framework to analyze extremal families under $ q $-dependence, generalizing Kleitman's classical result.
• Apply the stability result to the uniform case where all families are subsets of $ { [n] \choose k } $, proving near-optimal bounds for the Erd\'os Matching Conjecture.
• Establish a connection between $ q $-dependence and anti-Ramsey problems by linking the absence of rainbow matchings to structural constraints on color classes.
• Leverage double counting and extremal set-theoretic inequalities to derive tight bounds on the maximum family size under $ q $-dependence.
• Use the stability result to determine the exact value of $ ar(n,k,s) $ for $ n \geq sk + (s-1)(k-1) $ and $ k \geq 3 $.

Experimental results

Research questions

• RQ1 What is the maximum possible sum of sizes of $ s $ set families on $[n]$ that are $ q $-dependent for all $ n \geq q $ and $ s \geq 2 $?
• RQ2 How can a Hilton-Milner-type stability theorem be established for the Erd\'os Matching Conjecture in a wide range of parameters?
• RQ3 What is the minimum number of colors $ ar(n,k,s) $ required to color the $ k $-element subsets of $[n]$ without a rainbow matching of size $ s $?
• RQ4 Can the stability framework for $ q $-dependent families be used to resolve anti-Ramsey-type problems?
• RQ5 What structural constraints on set families ensure the absence of small union systems with disjoint sets?

Key findings

• The maximum sum $ \sum_{i=1}^s |\mathcal{F}_i| $ for $ q $-dependent families is determined exactly for all $ n \geq q $, $ s \geq 2 $, generalizing Kleitman's theorem.
• A Hilton-Milner-type stability result is established for the Erd\'os Matching Conjecture in the range $ n \geq (2+o(1))sk $, significantly improving earlier bounds by Bollob\'as, Daykin, and Erd\'os.
• The anti-Ramsey number $ ar(n,k,s) $ is determined exactly for all $ k \geq 3 $ and $ n \geq sk + (s-1)(k-1) $, resolving a problem posed by \'Ozkahya and Young.
• The stability framework allows the exact determination of extremal configurations under $ q $-dependence, revealing structural rigidity in near-optimal families.
• The results provide a unified approach to both the Erd\'os Matching Conjecture and anti-Ramsey problems via the $ q $-dependence condition.
• The paper establishes tight bounds on the size of families avoiding small unions of disjoint sets, with applications to rainbow matching avoidance.