Quantitative bounds for regular $3$-wise intersecting families

Frankston, Kahn and Narayanan proved that every regular increasing $3$-wise intersecting family of subsets of $[n]$ has cardinality $o(2^n)$ 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 $$\log \frac{2^n}{\mathrm{\mid }\mathcal{A}\mathrm{\mid }}\ge \frac{n}{2}{\left(\frac{\mathrm{\mid }\mathcal{A}\mathrm{\mid }}{2^n-\mathrm{\mid }\mathcal{A}\mathrm{\mid }}\right)}^2,$$ and consequently $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\le 2^n\sqrt{W(n)\mathrm{/}n}$, where $W$ is the principal Lambert function defined by $W(x)e^{W(x)}=x$ for $x\ge 0$. We also give a purely Fourier-analytic proof of the weaker estimate $$\mathrm{\mid }\mathcal{A}\mathrm{\mid }\le \frac{2^n}{1+n^{1\mathrm{/}3}}\mathrm{.}$$