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                * Princeton Discrete Math Seminar *
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Speaker: Maya Shankar (Stanford)

Thursday 5th September, 3:00 in Fine Hall 224.

Title: The Turán density of 4-uniform tight cycles

A tight cycle is an r-uniform hypergraph analogue of a cycle. Formally, the
length-k tight cycle C^(r)_k has k>r vertices v_1 ... v_k and edges v_{i+1}
... v_{i+r} for each residue i mod k.

We show that for all k sufficiently large and not divisible by 4, the Turán
density of the tight cycle C^(4)_k is exactly 1/2. That is, we show that the
densest 4-uniform hypergraph on n vertices with no copy of C^(4)_k has edge
density 1/2 + o(1) as n → ∞.

A key ingredient in the proof is a hypergraph analogue of the result that a
graph contains no odd cycles if and only if it is bipartite, which we hope
will be helpful towards understanding the Turán densities of other
cycle-like hypergraphs.

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