Minimum codegree threshold for $(K_4^3-e)$-factors

TL;DR

This paper determines the minimum codegree threshold for the existence of a (K_4^3-e)-factor in large 3-uniform hypergraphs, establishing an asymptotically optimal bound and conjecturing the exact threshold.

Contribution

It proves an asymptotically optimal minimum codegree condition for (K_4^3-e)-factors and conjectures the exact threshold, completing the asymptotic understanding for all F-factors on 4 vertices.

Findings

Established a threshold of (1/2 + γ)n for large hypergraphs.

Proved the bound is asymptotically best possible.

Formulated a conjecture for the exact threshold value.

Abstract

Given hypergraphs H and F, an F-factor in H is a spanning subgraph consisting of vertex disjoint copies of F. Let K_4^3-e denote the 3-uniform hypergraph on 4 vertices with 3 edges. We show that for \gamma>0 there exists an integer n_0 such that every 3-uniform hypergraph $H$ of order n > n_0 with minimum codegree at least (1/2+\gamma)n and 4|n contains a (K_4^3-e)-factor. Moreover, this bound is asymptotically the best possible and we further give a conjecture on the exact value of the threshold for the existence of a (K_4^3-e)-factor. Therefore, all minimum codegree thresholds for the existence of F-factors are known asymptotically for 3-uniform hypergraphs F on 4 vertices.