Sperner's Theorem F is an antichain.Then(). Lubell's proof (double counting) {1,2,3} maximal chain: {1.2} 1.3 23} 0cS1c·cSm-1C[nl of maximal chains in 21:n! {2} 3} ∀Sc[m, ☑ of maximal chains containing S:S!(n-S)! F is antichain 廿chain C,|FnC≤1 maximal chains crossings#all maximal chainsSperner’s Theorem F ￾ 2[n] is an antichain. Then |F| ⇥ ￾ n ￾n/2⇥ ⇥ . ∅ {1} {2} {3} {1,2} {1,3} {2,3} Lubell’s proof {1,2,3} (double counting) maximal chain: ⇤ ⇥ S1 ⇥ ··· ⇥ Sn￾1 ⇥ [n] # of maximal chains in 2[n]: n! {1,3} # of maximal chains containing S: F is antichain ∀ chain C, |F ⇤ C| ￾ 1 ⇥S ￾ [n], |S|!(n ￾ |S|)! # maximal chains crossing F ≤ # all maximal chains