The product measures of cross $t$-intersecting families

Abstract

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 $|{\mathcal{H}}_1||{\mathcal{H}}_2|\le (m^{n-t}{)}^2$. 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 $\min \left\{{\mu }_p({\mathcal{F}}_1),{\mu }_p({\mathcal{F}}_2)\right\}\le {\mu }_p(\mathcal{K}(n,t))$, where $\mathcal{K}(n,t)$ denotes the Katona family. This recovers an old result of Ahlswede--Katona.

Withdrawn with the agreement of all authors. This manuscript is being replaced by a substantially revised and extended paper that covers the remaining parameter cases and includes equality characterizations and dimension-free stability results. The new paper will be submitted separately to arXiv