The question under which minimum degree conditions a given hypergraph contains a Hamilton cycle has received a lot of attention over the past twenty years. In this talk, I will give an overview of the progress in the area and then present some new results.
I'll talk about a recent result which is joint work with Micha Christoph and Rajko Nenadov. In it, we show that if n is odd and p ≥ C log n/n, then with high probability Hamilton cycles in G(n, p) span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph G, that is, a graph G with odd n vertices and minimum degree n/2 + C for sufficiently large constant C, span its cycle space.
What is the minimum number of edges that a graph can have if it contains all n-vertex graphs of degeneracy D as subgraphs? We answer this question up to a polylogarithmic factor. In the talk I will explain the history of this and related problems and sketch our proof.
This is based on joint work with Peter Allen and Anita Liebenau.
The classical Andrásfai-Erdős-Sós Theorem states that for a certain constant δl>0 every n-vertex Kl+1-free graph with minimum degree greater than (l-1)n/l-δln must be l-partite. We will talk about a method for establishing the AES-type stability of an r-uniform hypergraph, and explore some of its applications.
In the 1980s Erdős and Sós introduced the notion of the uniform Turán density, which requires in addition to the classical notion of the Turán density, the edges of the host (hyper)graph to be distributed uniformly. Similarly, as the famous Tetrahedron Problem in the classical setting, determining the uniform Turán density of the complete 3-uniform 4-vertex hypergraph is a very challenging problem and has been open for more than 40 year. The first non-trivial hypergraph whose Turán density was determined is the broken tetrahedron (the 3-uniform 4-vertex hypergraph with 3-edges), which was shown to be 1/4 by Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. Currently, only constructions of 3-uniform hypergraphs with uniform Turán density equal to 0, 1/27, 4/27 and 1/4 are known. We expand this list by providing a sufficient condition on a 3-uniform hypergraph to have the uniform Turán density equal to 8/27 and we construct a hypergraph satisfying this condition.