Dimensions of hypergraphs

The dimension D(S) of a family S of subsets of n = (1,2, . . . . n> is defined as the minimum number of permutations of n such that every A E S is an intersection of initial segments of the permutations. Equivalent characterizations of D(S) are given in terms of suitable arrangements, interval dimension, order dimension, and the chromatic number of an associated hypergraph. We also comment on the maximum-sized family of k-element subsets of n having dimension m, and on the dimension of the family of all k-element subsets of n. The paper concludes with a series of alternative characterizations of D(S) = 2 and a list of open problems.