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 $μ_{\mathbf{p}}(\mathcal A)\le p_1$, where $μ_{\mathbf{p}}$ is the non-uniform product measure defined by $μ_{\mathbf{p}}(\mathcal{A})=\sum_{A\in\mathcal{A}} \prod_{i\in A} p_i \prod_{j\in [n]\setminus A}(1-p_j)$. 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 $μ_{\mathbf{p}}(\mathcal A)\le \prod_{i=1}^t p_i$. 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 $T\in \binom{[n]}{t}$ 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.