Improvement on the Erdős-Kleitman conjecture via the KKL theorem

In 1974, Erdős 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 $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\ge (1-2^{-(s-1)})\cdot 2^n\mathrm{.}$ 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 $(\frac{1}{2}+\epsilon )\cdot 2^n$ for some $\epsilon >0$ is a challenging problem. A breakthrough of Bucič, Letzter, Sudakov and Tran in 2018 showed that $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\ge (1-\frac{1}{s})\cdot 2^n$ via two very elegant and quite different approaches. Our main result shows that $$\mathrm{\mid }\mathcal{F}\mathrm{\mid }\ge (1-\frac{1}{s+(s-2)\frac{\log n}{2\sqrt{5}n}})\cdot 2^n$$ 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.