Cross-intersecting families of vectors
J\'anos Pach, G\'abor Tardos

TL;DR
This paper determines the maximum product of sizes for pairs of vector families with a cross-intersecting property, characterizes extremal pairs when the minimum sequence element exceeds a threshold, and discusses a conjecture for the other case.
Contribution
It generalizes and strengthens previous results on cross-intersecting families of vectors, providing a complete characterization for the case when the minimum element exceeds a certain bound.
Findings
Maximum product of sizes for r-cross-intersecting families determined.
Extremal pairs characterized for the case p_i > r+1.
Conjecture proposed and verified under additional assumptions for the other case.
Abstract
Given a sequence of positive integers , let denote the family of all sequences of positive integers such that for all . Two families of sequences (or vectors), , are said to be -cross-intersecting if no matter how we select and , there are at least distinct indices such that . We determine the maximum value of over all pairs of - cross-intersecting families and characterize the extremal pairs for , provided that . The case is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and answers a question of Zhang.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Analytic Number Theory Research · Mathematical Approximation and Integration
