A product inequality and its application to cross-intersecting families

Abstract

For an integer $r\ge 3$ and real numbers $a_1,\dots ,a_r\in (0,1)$, let $a=a_1\cdots a_r$. We show that \[ \prod_{i=1}^r(a_i+a_i^2+\cdots+a_i^r)\geq \prod_{i=1}^r(a_i+a_i^2+\cdots+a_i^{r-1}+a). \] This inequality enables us to bound the measure of $r$-cross $t$-intersecting families.

arXiv comment9 pages