Spanning spheres in Dirac hypergraphs

We show that an $n$-vertex $k$-uniform hypergraph, where all $(k-1)$-subsets that are supported by an edge are in fact supported by at least $n/2+o(n)$ edges, contains a spanning $(k-1)$-dimensional sphere. This generalises Dirac's theorem, and confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the absorption method or the regularity lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host hypergraph with a family of complete blow-ups.

This is joint work with Freddie Illingworth, Richard Lang, Olaf Parczyk, and Amedeo Sgueglia.