Towards extending the Ahlswede-Khachatrian theorem to cross t-intersecting families

TL;DR

This paper extends the Ahlswede-Khachatrian theorem to cross t-intersecting families, identifying the maximum weighted pairs and characterizing their structure for large t within specific probability ranges.

Contribution

It proves the maximum p-weight for cross t-intersecting families when r=1 and characterizes the unique optimal families for large t within certain probability intervals.

Findings

Maximum p-weight achieved by specific families for large t

Optimal families are isomorphic to ^t_1 in the specified range

Uniqueness of optimal families except at range endpoints

Abstract

Ahlswede and Khachatrian's diametric theorem is a weighted version of their complete intersection theorem, itself an extension of the $t$-intersecting Erd\H{o}s-Ko-Rado theorem. Their intersection theorem says that the maximum size of a family of subsets of $[n] = \{1, \dots, n\}$, every pair of which intersects in at least $t$ elements, is the size of certain trivially intersecting families proposed by Frankl. We address a cross intersecting version of their diametric theorem. Two families $\mathcal{A}$ and $\mathcal{B}$ of subsets of $[n]$ are {\em cross $t$-intersecting} if for every $A \in \mathcal{A}$ and $B \in \mathcal{B}$, $A$ and $B$ intersect in at least $t$ elements. The $p$-weight of a $k$ element subset $A$ of $[n]$ is $p^{k}(1-p)^{n-k}$, and the weight of a family $\mathcal{A}$ is the sum of the weights of its sets. The weight of a pair of families is the product of the weights of the families. The maximum $p$-weight of a $t$-intersecting family depends on the value of $p$. Ahlswede and Khachatrian showed that for $p$ in the range $[\frac{r}{t + 2r - 1}, \frac{r+1}{t + 2r + 1}]$, the maximum $p$-weight of a $t$-intersecting family is that of the family $\mathcal{F}^t_r$ consisting of all subsets of $[n]$ containing at least $t+r$ elements of the set $[t+2r]$. In a previous paper we showed a cross $t$-intersecting version of this for large $t$ in the case that $r = 0$. In this paper, we do the same in the case that $r = 1$. We show that for $p$ in the range $[\frac{1}{t + 1}, \frac{2}{t + 3}]$ the maximum $p$-weight of a cross $t$-intersecting pair of families, for $t \geq 200$, is achieved when both families are $\mathcal{F}^t_1$. Further, we show that except at the endpoints of this range, this is, up to isomorphism, the only pair of $t$-intersecting families achieving this weight.