Seminar

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I'll discuss some recent results on the Turán density of long cycle-like hypergraphs. Several years ago, Kamčev–Letzter–Pokrovskiy determined the Turán density of all sufficiently long 3-uniform tight cycles. Using a similar framework, Balogh–Luo determined the Turán density of sufficiently long 3-uniform tight cycles minus one edge (generalized to all lengths by Lidický–Mattes–Pfender using flag algebras) and I then did the same for sufficiently long 4-uniform tight cycles.

In this talk, I would like to discuss some of the ideas behind these results, aiming to outline a general strategy to prove similar results for tight cycles in larger uniformities or for other "cycle-like" hypergraphs. One key ingredient in several of these results, which I hope to prove in full, is a hypergraph analogue of the statement that a graph has no odd closed walks if and only if it is bipartite. More precisely, for various classes C of "cycle-like" r-uniform hypergraphs — including, for any k, the family C_k^(r) of r-uniform tight cycles of length k modulo r — we show that the C-homomorphically-free r-uniform hypergraphs are exactly those admitting a certain type of coloring of (r-1)-tuples of vertices. This provides a common generalization of results due to Kamčev–Letzter–Pokrovskiy and Balogh–Luo.