Abstract
In this paper, we consider an analog of the well-studied extremal problem for triangle-free subgraphs of graphs for uniform hypergraphs. A loose triangle is a hypergraph T consisting of three edges e, f and g such that \(|e \cap f| = |f \cap g| = |g \cap e| = 1\) and \(e \cap f \cap g = \emptyset \). We prove that if H is an n-vertex r-uniform hypergraph with maximum degree \(\triangle \), then as \(\triangle \rightarrow \infty \), the number of edges in a densest T-free subhypergraph of H is at least
For \(r = 3\), this is tight up to the o (1) term in the exponent. We also show that if H is a random n-vertex triple system with edge-probability p such that \(pn^3\rightarrow \infty \) as \(n\rightarrow \infty \), then with high probability as \(n \rightarrow \infty \), the number of edges in a densest T-free subhypergraph is
We use the method of containers together with probabilistic methods and a connection to the extremal problem for arithmetic progressions of length three due to Ruzsa and Szemerédi.
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This research is partially supported by the NSF Graduate Research Fellowship DGE-1650112; and partially supported by NSF FRG award DMS-1952786.
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This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1650112. Research partially supported by NSF FRG award DMS-1952786.
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Nie, J., Spiro, S. & Verstraëte, J. Triangle-Free Subgraphs of Hypergraphs. Graphs and Combinatorics 37, 2555–2570 (2021). https://doi.org/10.1007/s00373-021-02388-5
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DOI: https://doi.org/10.1007/s00373-021-02388-5