Spanning spheres in Dirac hypergraphs

TL;DR

This paper proves that under certain degree conditions, a k-uniform hypergraph contains a spanning subgraph homeomorphic to a sphere, extending Dirac's theorem topologically without using traditional complex methods.

Contribution

It introduces a new topological extension of Dirac's theorem for hypergraphs, confirming a conjecture with a novel proof avoiding standard complex techniques.

Findings

Hypergraphs with specified degree conditions contain a spherical spanning subgraph.

The proof avoids the Absorption Method, Regularity Lemma, and Blow-up Lemma.

Uses a new framework based on covering vertices with complete blow-ups.

Abstract

We show that a $k$-uniform hypergraph on $n$ vertices has a spanning subgraph homeomorphic to the $(k - 1)$-dimensional sphere provided that $H$ has no isolated vertices and each set of $k - 1$ vertices supported by an edge is contained in at least $n/2 + o(n)$ edges. This gives a topological extension of Dirac's theorem and asymptotically confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the Absorption Method, the Regularity Lemma or the Blow-up Lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host graph with a family of complete blow-ups.