Turán Problems For Tight Cycles

I will discuss two recent results regarding hypergraphs avoiding tight cycles of a fixed length. The main tool, 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 of tight cycles of length k modulo r — we show that C-hom-free hypergraphs are exactly those admitting a certain type of coloring of (r-1)-tuples of vertices.

This characterization allows us to reframe Turán-type problems as coloring problems. I will describe two applications of this approach: we compute the Turán density of sufficiently long 4-uniform tight cycles and provide upper bounds on the codegree Turán density of long tight cycles in all uniformities.

Some results are joint with Jozsef Balogh and Haoran Luo.

For information about the Richard P. Stanley Seminar in Combinatorics, visit… https://math.mit.edu/combin/