Abstract

Dirac’s theorem states that any n-vertex graph with minimum degree at least n/2 contains a Hamilton cycle. Rödl, Rucinski and Szemerédi showed that asymptotically the same bound gives a tight Hamilton cycle in any k-uniform hypergraph, where in this case “minimum degree” is interpreted as the minimum codegree, i.e. the minimum over all (k-1)-sets of the number of ways to extend that set to an edge. The notion of a tight cycle can be generalised to an l-cycle for any l at most k, and corresponding results for l-cycles were proved independently by Keevash, Kühn, Mycroft and Osthus and by Hàn and Schacht, and extended to the full range of l by Kühn, Mycroft and Osthus. However, l-cycles are essentially one-dimensional structures. A natural topological generalisation of Hamilton cycles in graphs to higher-dimensional dtructures is to ask for a spanning triangulation of a sphere in a 3-uniform hypergraph. We give an asymptotic Dirac-type result for this problem. Joint work with Agelos Georgakopoulos, Richard Montgomery and Bhargav Narayanan.