Speaker: Simon Piga
Title: Hypergraphs with arbitrarily small codegree Turán density
Abstract: Let k be at least 3. Given a k-uniform hypergraph H, the minimum codegree is the largest natural number d such that every (k-1)-set of V(H) is contained in at least d edges. Given a k-uniform hypergraph F, the codegree Turán density \gamma(F) of F is the smallest \gamma between 0 and 1 such that every k-uniform hypergraph on n vertices with minimum codegree at least (\gamma + o(1))n contains a copy of F. As in other variants of the hypergraph Turán problem, determining the codegree Turán density of a hypergraph is in general notoriously difficult and only few results are known.
We show that for every positive \epsilon, there is a k-uniform hypergraph F with 0<\gamma(F)<\epsilon. This is in contrast to the classical Turán density, which cannot take any value in the interval (0,k!/k^k) due to a fundamental result by Erdos.
