Sperner's Theorem F)is an antichain.Then(2). Lubell's proof (double counting) {1,23} maximal chain: {1,2} 13}23y 0cS1C…cSn-1c[ml of maximal chains in 21:n! {2} 3} ∀SC[ml, ☑ of maximal chains containing S:S!(n-S)! F is antichain >Vchain C,IFncl≤1 ∑1S1(n-1S1)!≤nd S∈FSperner’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|)! ￾ S￾F |S|!(n ￾ |S|)! ⇥ n!