The product measures of cross $t$-intersecting families

We investigate the product measures of intersection problems in extremal combinatorics. Invoking a recent result of He–Li–Wu–Zhang, we prove that for any $n\ge t\ge 3$ 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 ${\mu }_{p_1}({\mathcal{F}}_1){\mu }_{p_2}({\mathcal{F}}_2)\le (p_1{p}_2{)}^t$. Secondly, we study the intersection problems for integer sequences by proving that if ${\mathcal{H}}_1,{\mathcal{H}}_2\subseteq [m{]}^n$ are cross $t$-intersecting with $m>t+1$, then $\mathrm{\mid }{\mathcal{H}}_1\mathrm{\mid }\mathrm{\mid }{\mathcal{H}}_2\mathrm{\mid }\le (m^{n-t}{)}^2$. These results confirm two classical conjectures of Tokushige. As an application, we strengthen a recent theorem of Frankl–Kupavskii, generalizing the well-known IU-Theorem. Finally, we show that if $p\ge \frac{1}{2}$ and ${\mathcal{F}}_1,{\mathcal{F}}_2\subseteq 2^{[n]}$ are cross $t$-intersecting families, then $\min \{{\mu }_p({\mathcal{F}}_1),{\mu }_p({\mathcal{F}}_2)\}\le {\mu }_p(\mathcal{K}(n,t))$, where $\mathcal{K}(n,t)$ denotes the Katona family. This recovers an old result of Ahlswede–Katona.