The Number of Triple Systems Without Even Cycles

Abstract

For k ⩾ 4, a loose k-cycle Ck is a hypergraph with distinct edges e1, e2, …, ek such that consecutive edges (modulo k) intersect in exactly one vertex and all other pairs of edges are disjoint. Our main result is that for every even integer k ⩾ 4, there exists c > 0 such that the number of triple systems with vertex set [n] containing no Ck is at most \(2^{cn^2}\). An easy construction shows that the exponent is sharp in order of magnitude.

Our proof method is different than that used for most recent results of a similar flavor about enumerating discrete structures, since it does not use hypergraph containers. One novel ingredient is the use of some (new) quantitative estimates for an asymmetric version of the bipartite canonical Ramsey theorem.

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Acknowledgment

We are very grateful to Rob Morris for clarifying some technical parts of the proof in [44] at the early stages of this project. After those discussions, we realized that the method of hypergraph containers would not apply easily to prove Theorem 2 and we therefore developed new ideas. We are also grateful to József Balogh for providing us with some pertinent references, and to Jie Han for pointing out that our proof of Theorem 2 applies for r > 3 and odd k. Finally, many thanks to all referees for their helpful suggestions on the presentation of this paper. After their suggestions, the manuscript has undergone an extensive rewriting.

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Correspondence to Dhruv Mubayi.

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Research partially supported by NSF grant DMS-1300138.

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Mubayi, D., Wang, L. The Number of Triple Systems Without Even Cycles. Combinatorica 39, 679–704 (2019). https://doi.org/10.1007/s00493-018-3765-6

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Mathematics Subject Classification (2010)

  • 05A16
  • 05C55