Uniform s-cross-intersecting families
Peter Frankl, Andrey Kupavskii

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 -element subsets of the set in which any two sets intersect, has cardinality at most . We say that two non-empty families are are {\it -cross-intersecting}, if for any we have . In this paper we determine the maximum of for all . This generalizes a result of Hilton and Milner, who determined the maximum of for nonempty -cross-intersecting families.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
