The Suda-Tanaka-Tokushige conjecture for $\mathbf{p}$-biased intersecting families

Abstract

In 2017, Suda, Tanaka and Tokushige conjectured that if $1>p_1\ge \cdots \ge p_n>0$ with $p_3\le \frac{1}{2}$, then every intersecting family $\mathcal{A}\subseteq 2^{[n]}$ satisfies ${\mu }_{\mathbf{p}}(\mathcal{A})\le p_1$, where ${\mu }_{\mathbf{p}}$ is the non-uniform product measure defined by ${\mu }_{\mathbf{p}}(\mathcal{A})={\sum }_{A\in \mathcal{A}}{\prod }_{i\in A}{p}_i{\prod }_{j\in [n]\setminus A}(1-p_j)$. In addition, if $p_1>p_3$ or $p_1<\frac{1}{2}$, then equality holds if and only if $\mathcal{A}$ is a star centered at some $i\in [n]$ with $p_i=p_1$. In this paper, we prove this conjecture in the following stronger $t$-intersecting form: for any $t\ge 1$, if $p_{t+2}\le \frac{1}{t+1}$, then every $t$-intersecting family $\mathcal{A}\subseteq 2^{[n]}$ satisfies ${\mu }_{\mathbf{p}}(\mathcal{A})\le {\prod }_{i=1}^t{p}_i$. Moreover, when $p_{t+2}<\frac{1}{t+1}$, equality holds if and only if $\mathcal{A}=\{A\subseteq [n]:T\subseteq A\}$ for some $T\in \left(\genfrac{}{}{0ex}{}{[n]}{t}\right)$ with ${\prod }_{i\in T}{p}_i={\prod }_{i=1}^t{p}_i$. Our result unifies and generalizes the classical theorems of Fishburn-Frankl-Freed-Lagarias-Odlyzko and Friedgut.

12 pages