On the order dimension of 1-sets versus k-sets

A family F of s-subsets of [t] is a (θ, s, t)-family iff the intersection of any two distinct elements of F has cardinality less than θ. Let f(θ, s, t) be the greatest integer n such that there exists an (θ, s, t)-family of cardinality n. Let dim(1, k; n) denote the dimension of B_n(1, k), the suborder of the Boolean lattice on [n] consisting of 1-subsets and k-subsets of [n]. We use upper and lower bounds on f(θ, s, t) to derive new lower and upper bounds on dim(1, k; n). In particular we answer a question of Trotter by showing that dim(1, log n; n) = Ω(log³ n/log log n). The estimation of dim(1, log n; n) plays a critical role in the determination of the maximum dimension of an ordered set with fixed maximum degree. Previously it was only known that (log² n/4 < dim(1, log n; n) <log³ n.