Extremal Combinatorics "How large or how small a collection of finite objects can be,if it has to satisfy certain restrictions." graphs set systems(hypergraphs)
Extremal Combinatorics “How large or how small a collection of finite objects can be, if it has to satisfy certain restrictions.” graphs set systems (hypergraphs)
Set Systems(Families) family:F2 ground set: [n] F has a particular property >？≤F|≤？
Set Systems (Families) F 2[n] family: ground set: [n] F has a particular property ? |F| ?
Antichains F2 is an antichain {1,2,3} VA,B∈F,AEB 《1.213}23} (is antichain } {2 3} largest size:(m2l） “Is this the largest size for all antichains?
Antichains ∅ {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3} {1,2} {1,3} {2,3} ⌅A, B ⇥ F, A ⇤ B F 2 is an antichain [n] [n] k ⇥ is antichain largest size: n n/2⇥ ⇥ “Is this the largest size for all antichains?
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 # $#+ $'# $## - $* $$ " ! " 0!$ '$&' ' ('$# * 6 $ 34 8" # $ $$$ ! ! " " ! " 0 5 '"$ $ $ #$ 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)
Sperner's proof {1,2,3} {1,2,3} {1,2} {1,3} {2,3} {1,2} {13} 23} 1 {2} 3} 1) 2} 3) 0 0
Sperner’s proof ∅ {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3} ∅ {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}