On multipartite posets

A poset $\mathbf{P}=(X,⪯)$ is {\em $m$-partite} if $X$ has a partition $X=X_1\cup \mathrm{.}\mathrm{.}\mathrm{.}\cup X_m$ such that (1) each $X_i$ forms an antichain in $\mathbf{P}$, and (2) $x\prec y$ implies $x\in X_i$ and $y\in X_j$ where $i<j$. In this article we derive a tight asymptotic upper bound on the order dimension of $m$-partite posets in terms of $m$ and their bipartite sub-posets in a constructive and elementary way.