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New Results on Simplex-Clusters in Set Systems

Abstract

A d-simplex is defined to be a collection A1,..., Ad+1 of subsets of size k of [n] such that the intersection of all of them is empty, but the intersection of any d of them is non-empty. Furthemore, a d-cluster is a collection of d+1 such sets with empty intersection and union of size ≤ 2k, and a d-simplex-cluster is such a collection that is both a d-simplex and a d-cluster. The Erdös-Chvátal d-simplex Conjecture from 1974 states that any family of k-subsets of [n] containing no d-simplex must be of size no greater than (n-1/n-1). In 2011, Keevash and Mubayi extended this conjecture by hypothesizing that the same bound would hold for families containing no d-simplex-cluster. In this paper, we resolve Keevash and Mubayi’s conjecture for all 4 ≤ d + 1 ≤ k and n ≥ 2k - d + 2, which in turn resolves all remaining cases of the Erdös-Chvatal Conjecture except when n is very small (i.e. n < 2k-d + 2).

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Acknowledgements

I would like to thank the Pomona College research circle for introducing me to problems on clusters, and in particular to Shahriar Shahriari and Archer Wheeler for working with me in the early stages. I would also like to thank my advisors József Solymosi and Richard Anstee for their support, and in particular to Richard Anstee for many hours spent helping me edit. Finally, I would like to thank Péter Frankl for helpful comments and for pointing out additional references, and to the referees for a careful reading and helpful comments.

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Correspondence to Gabriel Currier.

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Currier, G. New Results on Simplex-Clusters in Set Systems. Combinatorica 41, 495–506 (2021). https://doi.org/10.1007/s00493-020-4441-1

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Mathematics Subject Classification (2010)

  • 05D05