On multipartite posets

Let m ≥ 2 be an integer. We say that a poset P = (X, ) is m-partite if X has a partition X = X1 ∪ · · · ∪Xm such that (1) each Xi forms an antichain in P, and (2) x ≺ y implies x ∈ Xi and y ∈ Xj where i, j ∈ {1, . . . , m} and i < j. If P is m-partite for some m ≥ 2, then we say it is multipartite. – In this article we discuss the order dimension of multipartite posets in general and derive tight asymptotic upper bounds on the order dimension of them in terms of their bipartite sub-posets. 2000 MSC: 06A05, 06A06, 06A07