# A Stability Theorem for Matchings in Tripartite 3-Graphs

## TL;DR

This paper proves a stability theorem for matchings in regular tripartite 3-graphs, showing that near-extremal structures are close to the unique extremal configuration, and addresses a related open question.

## Contribution

It establishes a stability version of the known matching bound in regular tripartite hypergraphs and answers a question on hypergraphs with general degree conditions.

## Key findings

- Regular tripartite hypergraphs with near-maximal matchings resemble the extremal configuration.
- The stability bound is explicit and quantifies structural closeness.
- The paper resolves an open question about matchings under broader degree conditions.

## Abstract

It follows from known results that every regular tripartite hypergraph of positive degree, with $n$ vertices in each class, has matching number at least $n/2$. This bound is best possible, and the extremal configuration is unique. Here we prove a stability version of this statement, establishing that every regular tripartite hypergraph with matching number at most $(1 + \varepsilon)n/2$ is close in structure to the extremal configuration, where "closeness" is measured by an explicit function of $\varepsilon$. We also answer a question of Aharoni, Kotlar and Ziv about matchings in hypergraphs with a more general degree condition.