Combinatorics Seminar

We prove that there exists an integer L_0 such that for every L>L_0 not divisible by 4, the tight cycle C^(4)_L has Turán density 1/2. Moreover, the extremal example differs from the complete oddly bipartite hypergraph in o(n^4) edges.
A key ingredient in our proof is an equivalent characterization (for each r and k) of the r-uniform hypergraphs that homomorphically avoid tight cycles of length k modulo r, in terms of colorings of (r-1)-tuples of vertices. This generalizes the result that a graph avoids all odd closed walks if and only if it is bipartite, answering a question of Kamčev, Letzter, and Pokrovskiy. In fact, our characterization applies to a far broader class of families than r-uniform tight cycles of length k mod r, hopefully bringing the Turán densities of many other "long-cycle-like" hypergraphs within reach.