On the Size of K-Cross-Free Families

Abstract

Two subsets A,B of an n-element ground set X are said to be crossing, if none of the four sets AB, A\B, B\A and X\(AB) are empty. It was conjectured by Karzanov and Lomonosov forty years ago that if a family F of subsets of X does not contain k pairwise crossing elements, then |F|=O(kn). For k=2 and 3, the conjecture is true, but for larger values of k the best known upper bound, due to Lomonosov, is |F|=O(knlogn). In this paper, we improve this bound for large n by showing that |F|=Ok(nlog*n) holds, where log* denotes the iterated logarithm function.

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Correspondence to Andrey Kupavskii.

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Research partially supported by Swiss National Science Foundation grants no. 200020-162884 and 200021-175977.

Research supported by the grant RNF 16-11-10014.

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Kupavskii, A., Pach, J. & Tomon, I. On the Size of K-Cross-Free Families. Combinatorica 39, 153–164 (2019). https://doi.org/10.1007/s00493-017-3792-8

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Mathematics Subject Classification (2010)

  • 05D05
  • 05C62