AK-type stability theorems on cross t-intersecting families

Abstract

Two families, $\mathcal{A}$ and $\mathcal{B}$, of subsets of $[n]$ are cross $t$-intersecting if for every $A\in \mathcal{A}$ and $B\in \mathcal{B}$, $A$ and $B$ intersect in at least $t$ elements. For a real number $p$ and a family $\mathcal{A}$ the product measure ${\mu }_p(\mathcal{A})$ is defined as the sum of $p^{|A|}(1-p{)}^{n-|A|}$ over all $A\in \mathcal{A}$. For every non-negative integer $r$, and for large enough $t$, we determine, for any $p$ satisfying $\frac{r}{t+2r-1}\le p\le \frac{r+1}{t+2r+1}$, the maximum possible value of ${\mu }_p(\mathcal{A}){\mu }_p(\mathcal{B})$ for cross $t$-intersecting families $\mathcal{A}$ and $\mathcal{B}$. In this paper we prove a stronger stability result which yields the above result.

24 pages, 4 figures