Perfect tilings of 3-graphs with the generalised triangle

TL;DR

This paper determines the exact minimum codegree threshold needed for perfect tilings of 3-uniform hypergraphs with a specific small hypergraph called the generalised triangle, advancing understanding of hypergraph tiling conditions.

Contribution

It establishes the best-possible minimum codegree condition for perfect tilings with the generalised triangle in 3-uniform hypergraphs and extends results to the rainbow tiling variant.

Findings

Established the exact minimum codegree threshold for perfect tilings.

Provided asymptotically optimal conditions for rainbow tilings.

Enhanced understanding of hypergraph tiling thresholds.

Abstract

We establish a best-possible minimum codegree condition for the existence of a perfect tiling of a $3$-uniform hypergraph $H$ with copies of the generalised triangle $T$, which is the 3-uniform hypergraph with five vertices $a, b, c, d, e$ and three edges $abc$, $abd$, $cde$. We also give an asymptotically-optimal minimum codegree condition for the rainbow version of the problem.