Sperner's Theorem Theorem(Sperner 1928) F C 2 is an antichain. (2) Emanuel Sperner (1905-1980) '$# $"$"$ ($'' ($ 2 $$#$'# ($* 0 )$""#"$#$*# 2 '$ .% .$0"#$ 1 ''$" '$'$$$ ' - $$"*#' ($$#0 ('+ ' ($$#'+ ($'"$ $$#0 3/4 ,'$#$!$$"##" 2 - "$ 0 $##$$'# ( $ $$$'# 0 )$ $#$$$7 #$#+$'# $## -$* $$" ! " 0!$'$&'' ('$#*6$34 8"#$ $$$ ! ! " " ! " 05'"$$$ #$0 ! ! " " ! '$##"+$ (##+ ($ (#$'#$0 ($$ ($$$0 "$ " ≤ ! ! " " 0!$" ( ∅ ⊂ ⊂ ⊂ ⊂ +" 0"'$$$7 #$#+" ($ $$ ($ (#' +$$'$ $$$'$ +$'# 0 &+$ ∈ "$!"'$$$ 0$$0 ' ∅ "$ $#' (+$$ ' "$ $'$#'0 $ #'+ ($##$$#! " $$# − $0 $$$ $" $ + $$$0 '#+# # (' ( - 0 0##"'$' ($ $''' ( $ − $&$&' ( $0 |F| n ⇤n/2⌅ ⇥ . Theorem (Sperner 1928) |F| n ⇤n/2⌅ ⇥ . Theorem (Sperner 1928) F 2[n] is an antichain. Sperner’s Theorem Emanuel Sperner (1905 - 1980)