  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.
