The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: A B;C D. The 3-crown consists of six distinct elements with the following property: A a subset of B, D and B a subset of C E and A a subset of C, F. A P-free family in the n-dimensional Boolean lattice is a subposet such that no collection of elements form the poset P. Note that the posets are not induced and may contain additional relations. There is a diamond-free family in the n-dimensional Boolean lattice of size (2+o(1))*( n choose n/2).

In this dissertation, we prove that any diamond-free family in the n-dimensional Boolean lattice has size at most (2:25 + o(1))

*( n choose 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.

There is a 3-crown-free family in the n-dimensional Boolean lattice of size n choose n/2. In this dissertation, we prove that any 3-crown-free family in the n-dimensional Boolean lattice has

size at most (2(3)^&half - 2)*( n choose n/2).