Large Cliques in Hypergraphs with Forbidden Substructures

Abstract

A result due to Gyárfás, Hubenko, and Solymosi (answering a question of Erdős) states that if a graph G on n vertices does not contain K2,2 as an induced subgraph yet has at least \(c\left(\begin{array}{c}n\\ 2\end{array}\right)\) edges, then G has a complete subgraph on at least \(\frac{c^2}{10}n\) vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced K2,2 which allows us to generalize their result to k-uniform hypergraphs. Our result also has an interesting consequence in discrete geometry. In particular, it implies that the fractional Helly theorem can be derived as a purely combinatorial consequence of the colorful Helly theorem.

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Acknowledgement

The author thanks Xavier Goaoc, Seunghun Lee, and two anonymous referees for pointing out some mistakes and making several other useful comments which greatly improved this manuscript.

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Correspondence to Andreas F. Holmsen.

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Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03930998).

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Holmsen, A.F. Large Cliques in Hypergraphs with Forbidden Substructures. Combinatorica 40, 527–537 (2020). https://doi.org/10.1007/s00493-019-4169-y

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Mathematics Subject Classification (2010)

  • 05C35
  • 05C65
  • 52A35