# On Perfect Matchings and tilings in uniform Hypergraphs

## TL;DR

This paper investigates Dirac-type problems in hypergraphs, establishing minimum codegree conditions for perfect matchings and tilings, and characterizing hypergraphs that contain such matchings.

## Contribution

It provides new minimum codegree thresholds for perfect matchings and tilings in uniform hypergraphs, including characterizations and asymptotic optimality results.

## Key findings

- Established minimum codegree thresholds for perfect matchings in hypergraphs.
- Proved conditions under which hypergraphs contain perfect tilings with specific subgraphs.
- Provided examples demonstrating the asymptotic sharpness of the thresholds.

## Abstract

In this paper we study some variants of Dirac-type problems in hypergraphs. First, we show that for $k\ge 3$, if $H$ is a $k$-graph on $n\in k\mathbb N$ vertices with independence number at most $n/p$ and minimum codegree at least $(1/p+o(1))n$, where $p$ is the smallest prime factor of $k$, then $H$ contains a perfect matching. Second, we show that if $H$ is a $3$-graph on $n\in 3\mathbb N$ vertices which does not contain any induced copy of $K_4^-$ (the unique $3$-graph with $4$ vertices and $3$ edges) and has minimum codegree at least $(1/3+o(1)))n$, then $H$ contains a perfect matching. Moreover, if we allow the matching to miss at most $3$ vertices, then the minimum degree condition can be reduced to $(1/6+o(1)))n$. Third, we show that if $H$ is a $3$-graph on $n\in 4\mathbb N$ vertices which does not contain any induced copy of $K_4^-$ and has minimum codegree at least $(1/8+o(1)))n$, then $H$ contains a perfect $Y$-tiling, where $Y$ represents the unique $3$-graph with $4$ vertices and $2$ edges. We also provide the examples showing that our minimum codegree conditions are asymptotically best possible. Our main tool for finding the perfect matching is a characterization theorem that characterizes the $k$-graphs with minimum codegree at least $n/k$ which contain a perfect matching.