Maya Sankar (IAS): Turán Problems For Tight Cycles

Abstract: 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, we show that r-uniform hypergraphs avoiding "tight closed walks" of length k modulo r are exactly those admitting a certain type of coloring of (r-1)-tuples of vertices. (In fact, we get a stronger result that applies to many other families of cycle-like hypergraphs.)

This characterization allows us to reframe problems about dense tight-cycle-free hypergraphs as coloring problems. I will highlight 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.

The second result is joint with Jozsef Balogh and Haoran Luo.