Sperners Theorem and its generalizations Matthias Beck Xueqin Wang - PDF document

Sperner’s Theorem and its generalizations Matthias Beck Xueqin Wang Thomas Zaslavsky SUNY Binghamton www.binghamton.edu/matthias

“Everything should be made as simple as possible, but not simpler.” Albert Einstein “The worst thing you can do to a problem is solve it completely.” Daniel Kleitman 2

Matroid theory Ehrhart theory Department Lattice! 3

Define a weak partial composition into p � � parts as an ordered p -tuple A 1 , . . . , A p such that A 1 , . . . , A p are pairwise disjoint sets. � � Theorem Suppose A j 1 , . . . , A jp for j = 1 , . . . , m are different weak set compositions into p parts with the condition that, for all 1 ≤ k ≤ p and all I ⊆ [ m ] with | I | = r +1 , there exist distinct i, j ∈ I such that either A ik = A jk or � � A ik ∩ A jl � = ∅ � = A jk ∩ A il . l � = k l � = k Then m 1 � ≤ r p � � | A j 1 | + ··· + | A jp | j =1 | A j 1 | ,..., | A jp | and m is bounded by the sum of the r p largest p -multinomial coefficients for inte- gers less than or equal to � � max | A j 1 | + · · · + | A jp | . 1 ≤ j ≤ m 4

Let S be an n -element set. Sperner’s Theorem (1928) Suppose A 1 , . . . , A m ⊆ S such that A k �⊆ A j for k � = j . Then � � n m ≤ . ⌊ n/ 2 ⌋ LYM Inequality (Lubell, Yamamoto, Meshalkin, 1960 ± 6) Suppose A 1 , . . . , A m ⊆ S such that A k �⊆ A j for k � = j . Then m 1 � � ≤ 1 . � n | A k | k =1 Both bounds can be attained for any n . 5

Theorem (Erd˝ os, 1945) Suppose { A 1 , . . . , A m } ⊆ P ( S ) contains no chains with r + 1 elements. Then m is bounded by the sum of the r largest bino- � n � mial coefficients , 0 ≤ k ≤ n . k Theorem (Rota–Harper, 1970) Suppose { A 1 , . . . , A m } ⊆ P ( S ) contains no chains with r + 1 elements. Then m 1 � � ≤ r . � n | A k | k =1 Both bounds can be attained for any n and r . 6

Theorem (Griggs–Stahl–Trotter, 1984) � � Suppose A j 0 , . . . , A jq are m different chains in P ( S ) such that A ji �⊆ A kl for all i and l and all j � = k . Then � � n − q m ≤ . ⌊ ( n − q ) / 2 ⌋ Theorem (Bollob´ as, 1965) � � Suppose A j , B j are m pairs of sets such that A j ∩ B j = ∅ for all j and A j ∩ B k � = ∅ for all j � = k . Then m 1 � � ≤ 1 . � | A j | + | B j | j =1 | A j | Both bounds can be attained for any n and q . 7

Define a weak composition of S into p parts � � as an ordered p -tuple A = A 1 , . . . , A p of sets A k such that A 1 , . . . , A p are pairwise disjoint and A 1 ∪ · · · ∪ A p = S . Theorem (Meshalkin, 1963) Suppose M = { A 1 , . . . , A m } is a class of weak compositions of S into p parts such that for all 1 ≤ k ≤ p the set { A j k } m j =1 forms an antichain. Then m = |M| is boun- ded by the largest p -multinomial coefficient for n . Theorem (Hochberg–Hirsch, 1970) 1 � � ≤ 1 . n � | A 1 | ,..., | A p | A ∈M Both bounds can be attained for any n and p . 8

Proof. p = 2: LYM. For general p , let M ( F ) = { ( A 2 , . . . , A p ) : ( F, A 2 , . . . , A p ) ∈ M} . Then 1 1 � � � = � n n � n −| A 1 | �� � | A 1 | ,..., | A p | A ∈M A ∈M | A 1 | | A 2 | ,..., | A p | 1 1 � � = � n � n −| F | � � | F | A ′ ∈M ( F ) F ∈M 1 | A 2 | ,..., | A p | where A ′ = ( A 2 , . . . , A p ), 1 � � · 1 ≤ � n | F | F ∈M 1 by the induction hypothesis, ≤ 1 by LYM. � 9

Sperner E GST M 10

Theorem Suppose M = { A 1 , . . . , A m } is a class of weak compositions of S into p parts such that for all 1 ≤ k < p the set { A j k : 1 ≤ j ≤ m } contains no chain of length r . Then 1 � ≤ r p − 1 � (a) n � | A 1 | ,..., | A p | A ∈M (b) m = |M| is bounded by the sum of the r p − 1 largest p -multinomial coefficients for n . 11

Projective geometry P n − 1 ( q ) Rank of a flat r ( a ) = dim a + 1 q -Gaussian coefficients n ! q � n � q = , k k ! q ( n − k )! q where n ! q = ( q n − 1)( q n − 1 − 1) · · · ( q − 1) Theorem (Rota–Harper, 1970) Suppose { a 1 , . . . , a m } ⊆ P n − 1 ( q ) contains no chains with r + 1 elements. m 1 � (a) ≤ r � � n j =1 r ( a j ) q (b) m is at most the sum of the r largest � n � Gaussian coefficients q for 0 ≤ j ≤ n j 12

A Meshalkin sequence of length p in P n − 1 ( q ) is a sequence a = ( a 1 , . . . , a p ) of flats whose join is ˆ 1 and whose ranks sum to n . If a is a Meshalkin sequence, we write r ( a ) = ( r ( a 1 ) , . . . , r ( a p )) for the sequence of ranks. For α = ( α 1 , . . . , α p ) we write � s 2 ( α ) = α i α j i<j and define the ( q -)Gaussian multinomial co- efficient as n ! q � n � q = . α α 1 ! q · · · α p ! q 13

Theorem Suppose M = { a 1 , . . . , a m } is a family of Meshalkin sequences of length p in P n − 1 ( q ) such that for all 1 ≤ k < p the set � � a j k : 1 ≤ j ≤ m contains no chain of length r . Then 1 q q s 2 ( r ( a )) ≤ r p − 1 � (a) � n � a ∈M r ( a ) (b) m = |M| is at most equal to the sum of the r p − 1 largest amongst the quantities � n � q q s 2 ( α ) α for α = ( α 1 , . . . , α p ) with all α k ≥ 0 and α 1 + · · · + α p = n 14

Corollary Suppose M = { a 1 , . . . , a m } is a family of Meshalkin sequences of length p in P n − 1 ( q ) such that for all 1 ≤ k < p the set � � a j k : 1 ≤ j ≤ m is an antichain. Then 1 � (a) q q s 2 ( r ( a )) ≤ 1 � n � a ∈M r ( a ) � n � q q s 2 ( α ) (b) m = |M| ≤ max α α (c) The bounds in (a) and (b) can be achieved for any n and p . 15

A “stranger” LYM inequality is Corollary Suppose M = { a 1 , . . . , a m } is a family of Meshalkin sequences of length p in P n − 1 ( q ) such that for all 1 ≤ k < p the set � a j � k : 1 ≤ j ≤ m contains no chain of length r . Then 1 � � n � a ∈M r ( a ) q is bounded by the sum of the r p − 1 largest expressions q s 2 ( α ) for α = ( α 1 , . . . , α p ) with all α k ≥ 0 and α 1 + · · · + α p = n . 16