Sparse hypergraphs with low independence number

Abstract

Let K (3)4 denote the complete 3-uniform hypergraph on 4 vertices. Ajtai, Erdős, Komlós, and Szemerédi (1981) asked if there is a function ω(d)→∞ such that every 3-uniform, K (3)4 -free hypergraph H with N vertices and average degree d has independence number at least \(\frac{N} {{d^{1/2} }}\omega (d)\). We answer this question by constructing a 3-uniform, K (3)4 -free hypergraph with independence number at most \(2\frac{N}{{{d^{1/2}}}}\). We also provide counterexamples to several related conjectures and improve the lower bound of some hypergraph Ramsey numbers.

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Correspondence to Dhruv Mubayi.

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Research supported in part by NSF grant DMS-1300138.

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Cooper, J., Mubayi, D. Sparse hypergraphs with low independence number. Combinatorica 37, 31–40 (2017). https://doi.org/10.1007/s00493-014-3219-8

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Mathematics Subject Classification (2000)

  • 05C65
  • 05D05
  • 05D10