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. is a diamond-free family in the n-dimensional Boolean lattice of size ( 2 − o ( 1 ) ) ( n ⌊ n / 2 ⌋ ) . In this paper, we prove that any diamond-free family in the n-dimensional Boolean lattice has size at most ( 2.25 + o ( 1 ) ) ( n ⌊ n / 2 ⌋ ) . 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.