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

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.