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

Dushnik and Mikr detirw tht: dimension of a partially crderc-d set X, denoted dim X, as the smallest positive integLi f for whkh thckre exist t linear estcnsions of X whose inter-saztion is the partial ordertng OF A'. Hiraguchi proved that ifr-t 2 2 and 1x1 G 2rt + 1, then dim X Q. n. dogart, Trotter and Kim& have gsven a forbidden c-• bposet characteriza?ion of Htraguchi " s Inequality by determining for c'at:h II 2 2, the mir lnum collection of pow5 (?,r surh that if 1x1 Ln + 1, the iiim X < n unless X co .tains one of the poscts from c,. Although &I = 24, for each II > 4, elr contains only the crown $fH-the poset consisting of all 1 element and It-i ekoment subsets of an ra element set ordered by inclusion. In this pape:, we consider a variar:t CJ~' dimension, caHeC interval dimension, and prove 3 forbidden subprlset chtiracerlzation c>f H;raguchi's inequality for interval dimension: If rt 2 2 and $2 d 2n-+ k, the interval dimension of X is less than n C:nless X con-tsms so,. " .