Dimension and Matchings in Comparability and Incomparability Graphs

We develop some new inequalities for the dimension of a finite poset. These inequalities are then used to bound dimension in terms of the maximum size of matchings. We prove that if the dimension of P is d and d ≥ 3, then there is a matching of size d in the comparability graph of P . There is no analogue of this result for cover graphs, as we show that there is a poset P of dimension d for which the maximum matching in the cover graph of P has size O(log d). On the other hand, there is a dual result in which the role of chains and antichains is reversed, as we show that there is also a matching of size d in the incomparability graph of P . The proof of the result for comparability graphs has elements in common with Perles’ proof of Dilworth’s theorem. Either result has the following theorem of Hiraguchi as an immediate corollary: dim(P ) ≤ |P |/2 when |P | ≥ 4.