In 3 dimensional matching, we are given a set $M\subseteq X\times Y\times Z$ where $|X|=|Y|=|Z|=n$. A matching in $M$ is a subset $T⊆M$ such that no elements in $T$ agree in any coordinate. The goal is to find a matching in $M$ of size $n$.
If we assume that $n\leq|M|=m<2n$, can we solve 3DM in polynomial-time?
I know that in a 2-bounded instance of 3DM, 3DM can be solved in polynomial-time. The 2-bounded instance is when no element appears in more than two triples in $M$. For $m<2n$, we do not necessarily have a 2-bounded instance.
Further, in a 3-bounded instance of 3DM, 3DM is NP-hard. The 3-bounded instance is when no element appears in more than three triples in $M$, that is $m\leq3n$.