An Erd\H os--Ko--Rado theorem for cross $t$-intersecting families

TL;DR

This paper proves a conjectured maximum product size for cross t-intersecting families of k-subsets in an n-set for large t, establishing uniqueness and stability of extremal families, and explores a weighted version of the problem.

Contribution

It verifies a conjecture on the maximum product of sizes of cross t-intersecting families for large t, including uniqueness and stability results, and introduces a weighted measure variant.

Findings

Maximum product of sizes is inom{n-t}{k-t}^2 for large t.

Extremal families are unique up to isomorphism.

Stability results show near-extremal families are close to the extremal structure.

Abstract

Two families $\mathcal{A}$ and $\mathcal{B}$, of $k$-subsets of an $n$-set, are {\em cross $t$-intersecting} if for every choice of subsets $A \in \mathcal{A}$ and $B \in \mathcal{B}$ we have $|A \cap B| \geq t$. We address the following conjectured cross $t$-intersecting version of the Erd\H os--Ko--Rado Theorem: For all $n \geq (t+1)(k-t+1)$ the maximum value of $|\mathcal{A}||\mathcal{B}|$ for two cross $t$-intersecting families $\mathcal{A}, \mathcal{B} \subset\binom{[n]}{k}$ is $\binom{n-t}{k-t}^2$. We verify this for all $t \geq 14$ except finitely many $n$ and $k$ for each fixed $t$. Further, we prove uniqueness and stability results in these cases, showing, for instance, that the families reaching this bound are unique up to isomorphism. We also consider a {\em $p$-weight} version of the problem, which comes from the product measure on the power set of an $n$-set.