Largest family without A ∪ B ⊆ C ∩ D
Journal of Combinatorial Theory Series A
On families of subsets with a forbidden subposet
Combinatorics, Probability and Computing
Combinatorics, Probability and Computing
Journal of Combinatorial Theory Series A
Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube
European Journal of Combinatorics
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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 (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.