Almost perfect matchings in $k$-partite $k$-graphs

TL;DR

This paper determines the minimum co-degree threshold for almost perfect matchings in k-partite k-graphs, extending previous results on perfect matchings and resolving an open question in the field.

Contribution

It establishes the minimum co-degree threshold for almost perfect matchings in k-partite k-graphs, generalizing prior work on perfect matchings.

Findings

Established the co-degree threshold for almost perfect matchings.

Extended the understanding of matchings in k-partite k-graphs.

Resolved an open question posed by R"odl and Rucinski.

Abstract

The minimum co-degree threshold for a perfect matching in a $k$-graph with $n$ vertices was determined by R\"odl, Ruci\'nski and Szemer\'edi for the case when $n\equiv 0\pmod k$. Recently, Han resolved the remaining cases when $n \not\equiv 0\pmod k$, establishing a conjecture of R\"odl, Ruci\'nski and Szemer\'edi. In this paper, we determine the minimum co-degree threshold for almost perfect matchings in $k$-partite $k$-graphs, answering a question of R\"odl and Ruci\'nski.