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Excluded subposets in the Boolean lattice

Abstract : We are looking for the maximum number of subsets of an n-element set not containing 4 distinct subsets satisfying $A ⊂B, C ⊂B, C ⊂D$. It is proved that this number is at least the number of the $\lfloor \frac{n }{ 2}\rfloor$ -element sets times $1+\frac{2}{ n}$, on the other hand an upper bound is given with 4 replaced by the value 2.
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Gyula O.H. Katona. Excluded subposets in the Boolean lattice. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. pp.229-230. ⟨hal-01184366⟩

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