Uniform s-cross-intersecting families

TL;DR

This paper extends classical intersection theorems by determining the maximum combined size of two families of k-element subsets of [n] that are s-cross-intersecting, generalizing previous results for the case s=1.

Contribution

It generalizes the Hilton-Milner theorem by establishing the maximum sum of sizes for s-cross-intersecting families for all n and s.

Findings

Determined maximum of ||+|| for s-cross-intersecting families.

Extended classical intersection results to s-cross-intersecting case.

Provided exact bounds for all parameters n, k, s.

Abstract

In this paper we study a question related to the classical Erd\H{o}s-Ko-Rado theorem, which states that any family of $k$-element subsets of the set $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most ${n-1\choose k-1}$. We say that two non-empty families are $\mathcal A, \mathcal B\subset {[n]\choose k}$ are {\it $s$-cross-intersecting}, if for any $A\in\mathcal A,B\in \mathcal B$ we have $|A\cap B|\ge s$. In this paper we determine the maximum of $|\mathcal A|+|\mathcal B|$ for all $n$. This generalizes a result of Hilton and Milner, who determined the maximum of $|\mathcal A|+|\mathcal B|$ for nonempty $1$-cross-intersecting families.