Maximum cardinality of a set of subsets
1
0
$begingroup$
Let $N$ be a system of subsets of the set $X = {1,2,3,cdots ,n }$ such that there are no three elements $A,B,C in N$ such that $A subset B subset C$ . Prove that $$|N| leq 2 cdot {{n}choose{ lfloor n/2 rfloor}}. $$ I have thought about the fact that a semi-independent system of subsets can have at most two elements in common with any chain from $(mathcal{P}(X), subseteq),$ but I do not know how to continue using Sperner's theorem.
combinatorics discrete-mathematics order-theory extremal-combinatorics
share | cite | improve this question
edited Jan 12 at 22:33
...