Discrete Mathematics seminar - Positive co-degree density of r-graphs

Abstract: In an r-uniform hypergraph H (which we will often call an r-graph for short), the co-degree of a set S of r-1 vertices is simply the number of hyperedges of H which contain S. The minimum co-degree of H is the minimum co-degree over all (r-1)-sets S, while the minimum positive co-degree of H is the largest integer k such that if an (r-1)-set S is contained in some hyperedge of H, then S is contained in at least k hyperedges of H.

Co-degree in r-graphs can be seen as a natural generalization of degree in 2-graphs, and several measures of extremality using co-degree have been considered. The best known of these is the co-degree Turán number coex(n,F), the largest possible minimum co-degree in an n-vertex, F-free r-graph. In this talk, inspired by the study of minimum co-degree problems, we shall introduce the positive co-degree Turán number, the largest possible minimum positive co-degree in an n-vertex, F-free r-graph. Our primary goals will be to motivate consideration of minimum positive co-degree problems and to give some general results on the (sometimes surprising) behavior of positive co-degree Turán numbers. We also highlight a number of open directions in this area.

This talk represents joint work with Cory Palmer and Nathan Lemons.

To receive the Zoom link, please contact Victor Falgas Ravry.