Let 2[n] denote the power set of [n]:={1,2,...,n}. A collection
\B\subset 2^{[n]} forms a d-dimensional {\em Boolean algebra} if there
exist pairwise disjoint sets X0β,X1β,...,Xdββ[n], all non-empty
with perhaps the exception of X0β, so that \B={X_0\cup \bigcup_{i\in I}
X_i\colon I\subseteq [d]}. Let b(n,d) be the maximum cardinality of a family
\F\subset 2^X that does not contain a d-dimensional Boolean algebra.
Gunderson, R\"odl, and Sidorenko proved that b(n,d)β€cdβnβ1/2dβ 2n where cdβ=10d2β21βdddβ2βd.
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\subseteq \tsupn is defined by h_n(\F):=\sum_{F\in \F}1/{{n\choose |F|}}.
We prove the following Tur\'an type theorem. If \F\subseteq 2^{[n]} contains
no d-dimensional Boolean algebra, then h_n(\F)\leq 2(n+1)^{1-2^{1-d}} for
sufficiently large n. This results implies b(n,d)β€Cnβ1/2dβ 2n, where C is an absolute constant independent of n and d. As a
consequence, we improve several Ramsey-type bounds on Boolean algebras. We also
prove a canonical Ramsey theorem for Boolean algebras.Comment: 10 page