Maximal dimensional partially ordered sets III: a characterization of Hiraguchi's inequality for interval dimension

Dushnik and Miller defined the dimensions of a partially ordered set X,denoted dim X, as the smallest positive integer t for which there exist t linear extensions of X whose intersection is the partial ordering on X. Hiraguchi proved that if n ≥2 and |X| ≤2n+1, then dim X ≤ n. Bogart, Trotter and Kimble have given a forbidden subposet characterization of Hiraguchi's inequality by determining for each n ≥ 2, the minimum collection of posets ?_n such that if |X| ?2n+1, the dim X < n unless X contains one of the posets from ?_n. Although |?₃|=24, for each n ≥ 4, ?_n contains only the crown S⁰_n — the poset consisting of all 1 element and n ? 1 element subsets of an n element set ordered by inclusion. In this paper, we consider a variant of dimension, called interval dimension, and prove a forbidden subposet characterization of Hiraguchi's inequality for interval dimension: If n ≥2 and |X 2n+1, the interval dimension of X is less than n unless X contains S⁰_n.