Quantitative bounds for regular $3$-wise intersecting families

Abstract: Frankston, Kahn and Narayanan proved that every regular increasing $3$-wise intersecting family of subsets of $[n]$ has cardinality $o(2<sup>n)$ using Friedgut's junta theorem. We give a short quantitative proof using elementary tools from the analysis of Boolean functions and entropy. More precisely, if $\mathcal{A}\subseteq\mathcal{P}_n$ is a nonempty $3$-wise intersecting family that is both regular and increasing, then $$\log\frac{2^{n}{|\mathcal{A}|}\ge} \frac{n}{2}\left(\frac{|\mathcal{A}|}{2^{n-|\mathcal{A}|}\right)^2,}$$ and consequently $|\mathcal{A}|\le 2<sup>n\sqrt{W(n)/n}$, where $W$ is the principal Lambert function defined by $W(x)e<sup>{W(x)}=x$ for $x\ge0$. We also give a purely Fourier-analytic proof of the weaker estimate $$|\mathcal{A}|\le \frac{2^{n}{1+n^{1/3}}.}$$