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 diamond-free 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 . In this paper, we prove that any diamond-free family in the n-dimensional Boolean lattice has size at most . Furthermore, we show that the so-called Lubell function of a diamond-free family in the n-dimensional Boolean lattice which contains the empty set is at most 2.25+o(1), which is asymptotically best possible.