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: $A\subset B,C\subset D$. 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 $(2-o(1)){n\choose\lfloor n/2\rfloor}$. In this paper, we prove that any diamond-free family in the $n$-dimensional Boolean lattice has size at most $(2.25+o(1)){n\choose\lfloor n/2\rfloor}$. Furthermore, we show that the so-called Lubell function of a diamond-free family in the $n$-dimensional Boolean lattice is at most $2.25+o(1)$, which is asymptotically best possible.