On diamond-free subposets of the Boolean lattice

The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: A ⊂ B,C ⊂ D. A diamondfree family in the n-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements B and C may or may not be related. There is a diamond-free family in the n-dimensional Boolean lattice of size (2−o(1)) ( n bn/2c ) . In this paper, we prove that any diamond-free family in the ndimensional Boolean lattice has size at most (2.25 + o(1)) ( n bn/2c ) . Furthermore, we show that the so-called Lubell function of a diamond-free family in the ndimensional Boolean lattice which contains the empty set is at most 2.25+o(1), which is asympotically best possible. AMS 2010 Subject Classification: Primary 06A07; Secondary 05D05, 05C35