Abstract
Given a sequence of positive integers \(p=(p_1,\dots ,p_n)\), let \(S_p\) denote the family of all sequences of positive integers \(x=(x_1,\ldots ,x_n)\) such that \(x_i\le p_i\) for all \(i\). Two families of sequences (or vectors), \(A,B\subseteq S_p\), are said to be \(r\) -cross-intersecting if no matter how we select \(x\in A\) and \(y\in B\), there are at least \(r\) distinct indices \(i\) such that \(x_i=y_i\). We determine the maximum value of \(|A|\cdot |B|\) over all pairs of \(r\)-cross-intersecting families and characterize the extremal pairs for \(r\ge 1\), provided that \(\min p_i>r+1\). The case \(\min p_i\le r+1\) 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, Borg, Frankl, Füredi, Livingston, Moon, and Tokushige, and answers a question of Zhang.
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Acknowledgments
We are indebted to G. O. H. Katona, R. Radoičić, and D. Scheder for their valuable remarks, and to an anonymous referee for calling our attention to the manuscript of Borg [5].
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J. Pach is supported by OTKA under ERC projects GraDR and ComPoSe 10-EuroGIGA-OP-003, and by Swiss National Science Foundation Grants 200020-144531 and 200021-137574. G. Tardos is supported by OTKA grant NN-102029, the “Lendület” project of the Hungarian Academy of Sciences and by EPFL.
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Pach, J., Tardos, G. Cross-Intersecting Families of Vectors. Graphs and Combinatorics 31, 477–495 (2015). https://doi.org/10.1007/s00373-015-1551-4
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DOI: https://doi.org/10.1007/s00373-015-1551-4