Tokushige's conjecture on measures of cross $t$-intersecting families
Abstract
We investigate the product measure version (also known as the $p$-weight version) of intersection problems in extremal combinatorics. Firstly, we prove that for any \(n \geq t \geq 3\) and \(p_1, p_2 \in (0, \frac{1}{t+1})\), if \(\mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}\) are cross \(t\)-intersecting families, then $\mu_{p_1}(\mathcal{F}_1)\mu_{p_2}(\mathcal{F}_2)\le (p_1p_2)^t$. This resolves a conjecture of Tokushige for \(t \geq 3\). Secondly, we study the intersection problems for integer sequences and prove that if $\mathcal{H}_1, \mathcal{H}_2 \subseteq [m]^{n}$ are cross $t$-intersecting with \(m > t+1\), then $|\mathcal{H}_1|| \mathcal{H}_2|\leq (m^{n-t})^2$. This confirms another conjecture of Tokushige for \(m > t+1\). As an application, we strengthen a recent theorem of Frankl and Kupavskii, generalizing the well-known IU-Theorem. Finally, we show that if \(p \geq \frac{1}{2}\) and $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $t$-intersecting families, then $\min \left\{\mu_{p}(\mathcal{F}_1),\mu_{p}(\mathcal{F}_2)\right\} \leq \mu_{p}(\mathcal{K}(n,t))$, where $\mathcal{K}(n,t)$ denotes the Katona family. This recovers an old result of Ahlswede and Katona.