AK-type stability theorems on cross t-intersecting families

TL;DR

This paper establishes stability theorems for cross t-intersecting families of subsets, determining maximum product measures under certain conditions and providing stronger stability results for these combinatorial structures.

Contribution

It introduces new stability theorems for cross t-intersecting families, extending classical results by quantifying how close families are to extremal configurations.

Findings

Maximum product measure determined for large t and specific p ranges.

Stability results show near-extremal families resemble extremal configurations.

Provides bounds and conditions for cross t-intersecting families' measures.

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 $\frac r{t+2r-1}\leq p\leq\frac{r+1}{t+2r+1}$, 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.