Boolean algebras and Lubell functions

Abstract

Let 2[n] denote the power set of [n], where [n]={1,2,…,n}. A collection B⊂2[n] forms a d-dimensional Boolean algebra if there exist pairwise disjoint sets X0,X1,…,Xd⊆[n], all non-empty with perhaps the exception of X0, so that B={X0∪⋃i∈IXi:I⊆[d]}. Let b(n,d) be the maximum cardinality of a family F⊂2X that does not contain a d-dimensional Boolean algebra. Gunderson, Rödl, and Sidorenko proved that b(n,d)≤cdn−1/2d⋅2n where cd=10d2−21−ddd−2−d.In this paper, we use the Lubell function as a new measurement for large families instead of cardinality. The Lubell value of a family of sets F with F⊆2[n] is defined by hn(F)=∑F∈F1/(n|F|). We prove the following Turán type theorem. If F⊆2[n] contains no d-dimensional Boolean algebra, then hn(F)≤2(n+1)1−21−d for sufficiently large n. This result implies b(n,d)≤Cn−1/2d⋅2n, where C is an absolute constant independent of n and d. With some modification, the ideas in Gunderson, Rödl, and Sidorenko's proof can be used to obtain this result. We apply the new bound on b(n,d) to improve several Ramsey-type bounds on Boolean algebras. We also prove a canonical Ramsey theorem for Boolean algebras.