Improvement on the Erd\H{o}s-Kleitman conjecture via the KKL theorem

TL;DR

This paper advances the Erd ext{"o}s-Kleitman conjecture by establishing a new lower bound on the size of families avoiding large matchings, using the KKL theorem and other combinatorial techniques.

Contribution

It introduces a novel bound improving previous results on the Erd ext{"o}s-Kleitman conjecture by leveraging the KKL theorem and linear algebra methods.

Findings

New lower bound on family size avoiding large matchings

Connection established between Erd ext{"o}s-Kleitman conjecture and KKL theorem

Independent weaker bound using linear algebra and combinatorics

Abstract

In 1974, Erd\H{o}s and Kleitman conjectured that if a family $\mathcal{F}\subseteq 2^{[n]}$ contains no matching of size \(s\) and is maximal with respect to this property, then $ |\mathcal{F}|\ge \left(1-2^{-(s-1)}\right)\cdot 2^{n}. $ For decades, the best general lower bound remained the trivial $2^{n-1}$. About a decade ago, Frankl and Tokushige emphasized that obtaining a bound of the form $\left(\frac{1}{2}+\varepsilon\right)\cdot 2^n$ for some $\varepsilon>0$ is a challenging problem. A breakthrough of Buci\v{c}, Letzter, Sudakov and Tran in 2018 showed that $ |\mathcal{F}|\ge \left(1-\frac{1}{s}\right)\cdot 2^n $ via two very elegant and quite different approaches. Our main result shows that $$ |\mathcal{F}|\ge \left( 1 - \frac{1}{s + (s-2)\frac{\log n}{2\sqrt{5}n}} \right)\cdot 2^n $$ by exploiting a connection to the cornerstone result of Kahn, Kalai and Linial on influences of Boolean functions. Independently, we can also obtain a weaker improvement combining the linear algebra method with a combinatorial twist.