Codegree thresholds for covering 3-uniform hypergraphs

TL;DR

This paper investigates the minimum codegree thresholds for covering 3-uniform hypergraphs with specific subgraphs, providing exact values for certain cases and bounds for others, advancing understanding in hypergraph covering problems.

Contribution

The paper determines the exact codegree threshold c_2(n, K_4) for large n and provides bounds for other subgraphs, introducing new results in hypergraph covering thresholds.

Findings

Exact threshold c_2(n, K_4) for n > 98.

Extremal configurations characterized for n > 998.

Bounds on c_2(n, F) for K_4^-, K_5^-, and C_5.

Abstract

Given two 3-uniform hypergraphs F and G, we say that G has an F-covering if we can cover V(G) by copies of F. The minimum codegree of G is the largest integer d such that every pair of vertices from V(G) is contained in at least d triples from E(G). Define c_2(n,F) to be the largest minimum codegree among all n-vertex 3-graphs G that contain no F-covering. This is a natural problem intermediate (but distinct) from the well-studied Tur\'an problems and tiling problems. In this paper, we determine c_2(n, K_4) (for n>98) and the associated extremal configurations (for n>998), where K_4 denotes the complete 3-graph on 4 vertices. We also obtain bounds on c_2(n,F) which are apart by at most 2 in the cases where F is K_4^- (K_4 with one edge removed), K_5^-, and the tight cycle C_5 on 5 vertices.