Order 11: 127-134, 1994. 127 © 1994 KtuwerAcademic Publishers. Printed in the Netherlands. The Dimension of Suborders of the Boolean Lattice G. R. BRIGHTWELL Department of Mathematics~ London School of Economics, Houghton Street, London WC2A 2AE, U.K. (E-mail: grbt O@phoenix.cambridge.ac.uk) H. A. KIERSTEAD Department of Mathemutics, Arizona State University~ Tempe, Arizona 85287, U.S.A. (E-mail: kierstead@muth.la.asu.edu) A. V. KOSTOCHKA Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk 90, Russia (E-mail: sasha@math.nsk.su) and W. T. TROTTER Bell Communications Research, 445 South Street 2L-367, Morristown, NJ 07962, U.S.A., and Department of Mathematics, Arizona State University, Tempe AZ 85287, U.S.A. (E-mail: wtt@bellcore.com) Communicated by W. Z Trotter (Received: 20 September 1993; accepted: 15 May 1994) Abstract. We consider the order dimension of suborders of the Boolean lattice/~n. In particular we show that the suborder consisting of the middle two levels of l~n has dimension at most 6 log3 n. More generally, we show that the suborder consisting of levels s and s + k of l~. has dimension O(k e log n). Mathematics Subject Classifications (1991). 06A07, 05C35. Key words. Ordered set, dimension, Bollean lattice, suborder. 1. Introduction For any positive integer n, let [n] = {1, 2,.,., n}, let B,~ be the collection of subsets of [n], and let/~,~ = (B,~, _C) denote the Boolean lattice, where the subsets of [n] are ordered by inclusion. For a finite set A, let C(A, k) denote the collection of k-element subsets of A. For integers n, s and t with 0 ~< s < t ~< n, let B,~(s, t) The research of the second author was supported by Office of Naval Research Grant N00014-90-J- 1206. The research of the third author was supported by Grant 93-011-1486 of the Russian Fundamental Research Foundation.