Estimating the Size of the Largest Families not Containing Tree-like - PowerPoint PPT Presentation

Estimating the Size of the Largest Families not Containing Tree-like Posets Wei-Tian Li Jerrold R. Griggs & Linyuan Lu Department of Mathematics University of South Carolina 24th Cumberland Conference May 14th 2011

Theorem (Sperner, 1928) Let A be an inclusion-free family of subsets of [ n ] . Then � n � |A| ≤ . ⌊ n 2 ⌋

Theorem (Sperner, 1928) Let A be an inclusion-free family of subsets of [ n ] . Then � n � |A| ≤ . ⌊ n 2 ⌋ Poset P = ( P , ≤ ) A poset P 1 = ( P 1 , ≤ 1 ) contains another poset P 2 = ( P 2 , ≤ 2) as a subposet if there exists an injection f from P 2 to P 1 such that f ( a ) ≤ 1 f ( b ) whenever a ≤ 2 b . Example: C t c �− → C b c P 2 P 1 t t ❅ � f : b �− → B B t ❅ � a t a �− → A A t

The Boolean lattice B n = (2 [ n ] , ⊆ ) is the poset consisting of the power set of [ n ] and the inclusion relation as the partial order. A full chain C in B n is a collection of n + 1 subsets as follows: ∅ ⊂ { a 1 } ⊂ { a 1 , a 2 } · · · ⊂ { a 1 , . . . , a n } . A P-free family F is a collection of subsets of [ n ] such that it does not contain P as a subsposet. The largest size of a P -free family of subsets of [ n ] is denoted by La ( n , P ). The sum of middle k binomial coefficients is denoted by Σ( n , k ) and B ( n , k ) is the family of subsets of middle k sizes.

r → t � �� � La ( n , P ) for various posets r .... r r r → ◗ ✑ � �� � r .... P ✏ ❅✑ ◗ � r ❍✏ P ✟ r r r r Th, 1998 k { r r r k − 2 { k ≥ 3 k { . . . r DK, 2007 . . . � n � . . . GL, 2009 ∼ k r ⌊ n r 2 ⌋ ✟P ✏ � n � r ✏ ❍ r P ∼ ( k − 1) r .... ⌊ n 2 ⌋ r r r r ր r � �� � r .... � �� � E, 1945 s r r r r ❙ ❆ ✁ ✓ r = Σ( n , k − 1) � �� � ... r , s ≥ 2 ❙ ❆ ✓ ✁ → r r ր Th, 1998 ❜ r ✧ r ❇ ✧✧ ❜ ❙ ✓ ✂ r r � n � ✡ ❜ ❏ ∼ ❇ ✡ ❏ ✂ ⌊ n ր 2 ⌋ ... ❙ ✓ S, 1928 r r DK, 2007 r → r r � n � ���� ∼ 2 ❅ � = Σ( n , 1) ⌊ n 2 ⌋ s ց KT, 1983 ❅ � � n � DKSw, 2005 ∼ D: De Bonis ⌊ n ց → 2 ⌋ 2 k � ❅ = Σ( n , 2) E: Erd˝ os r r � �� � ... G: Griggs r r r r r ◗ GL, 2009 ✓ ❙ ✄ ✄ ✄ ✄ K: Katona � � ◗ n r r ∼ L: Lu ⌊ n 2 ⌋ ✓ ❙ ❅ ✄ ✄ ✄ ◗ ✄ ... S: Sperner r r r r r r ❙ ✓ ❅ k ≥ 2 Sw: Swanepoel � �� � GL-L., 2011 GK, 2008 T: Tarj´ an ❙ ✓ ❅ 2 k � � � � ≤ 2 3 n n ∼ ⌊ n ⌊ n Th: Thanh r r r 11 2 ⌋ 2 ⌋ � n � Bukh 2010, All tree-posets P satisfy La ( n , P ) ∼ ( h ( P ) − 1) . ⌊ n 2 ⌋

Let F ⊂ 2 [ n ] . For each full chain C in B n , if a subset E ∈ F ∩ C , then we associate E and C together to get a pair ( E , C ). Count the number of pairs in two different ways: (1) There are | E | !( n − | E | )! full chains containing a set E ∈ F . (2) For each full chain C , it contains |F ∩ C| subsets in F .

Let F ⊂ 2 [ n ] . For each full chain C in B n , if a subset E ∈ F ∩ C , then we associate E and C together to get a pair ( E , C ). Count the number of pairs in two different ways: (1) There are | E | !( n − | E | )! full chains containing a set E ∈ F . (2) For each full chain C , it contains |F ∩ C| subsets in F . The Lubell function ¯ h ( F )(or ¯ h n ( F )) of F is defined to be the number of pairs ( E , C ) over n !. | E | !( n − | E | !) � ¯ h ( F ) = n ! E ∈F 1 � = (weighted sum of subsets) � n � | E | E ∈F = ave C |F ∩ C| (ave. no. of times C meet F )

Lemma Let F be a collection of subsets of [ n ] . If ¯ h ( F ) ≤ m, for some real number m > 0 , then � n � |F| ≤ m . ⌊ n 2 ⌋ Moreover, if m is an integer , then |F| ≤ Σ( n , m ) , and equality holds if and only if (1) F = B ( n , m ) (when n + m is odd), or � � n (2) F = B ( n , m − 1) together with any subsets of sizes ( n + m ) / 2 ( n ± m ) / 2 when n + m is even.

Proof. If ¯ h ( F ) ≤ m , then � n � � n � ⌊ n � � 2 ⌋ � ≤ m |F| = 1 ≤ . � n ⌊ n 2 ⌋ | E | E ∈F E ∈F If m is an integer, view ¯ h ( F ) as the weighted sum of the sets in F . q s ... s s s ( n + m ) / 2 s s s s s s s s s s s s s s s s s s s s s s s s s s s s s ⌊ n / 2 ⌋ s s s s s s s s s s s s s s s s s ... s ( n − m ) / 2 q

Given a poset P , let e ( P ) be the maximum m such that for all n , � [ n ] � the union of any m consecutive levels � m does not i =1 s + i contain P as a subposet. � [ n ] � [ n ] � � Example: For all n , the union of two levels ∪ , for k +1 k q q � ❅ since no two k -sets 0 ≤ k ≤ n − 1, contains no butterfly B = q q both contain two same ( k − 1)-subsets, so e ( B ) ≥ 2. However, the union of three consecutive levels must contain B when n ≥ 3. So e ( B ) = 2.

The limit La ( n , P ) lim � n � n →∞ ⌊ n 2 ⌋ is defined to be π ( P ). Conjecture (Griggs and Lu, 2009) For any finite poset, π ( P ) exists and is an integer.

The limit La ( n , P ) lim � n � n →∞ ⌊ n 2 ⌋ is defined to be π ( P ). Conjecture (Griggs and Lu, 2009) For any finite poset, π ( P ) exists and is an integer. fake line The family B ( n , e ( P )) contains no P , so when it exists, π ( P ) must be at least e ( P ). Observation (Saks and Winkler) All posets with π ( P ) determined satisfied e ( P ) = π ( P ).

Let λ n ( P ) be max ¯ h ( F ) over all P -free families F ⊂ 2 [ n ] . Suppose F is P -free and |F| = La ( n , P ). Then La ( n , P ) 1 � � = ¯ ≤ h ( F ) ≤ λ n ( P ) . � n � n � ⌊ n 2 ⌋ | E | E ∈F Define λ ( P ) = lim n →∞ λ n ( P ). We have e ( P ) ≤ π ( P ) ≤ λ ( P ) if both limits exist.

Let λ n ( P ) be max ¯ h ( F ) over all P -free families F ⊂ 2 [ n ] . Suppose F is P -free and |F| = La ( n , P ). Then La ( n , P ) 1 � � = ¯ ≤ h ( F ) ≤ λ n ( P ) . � n � n � ⌊ n 2 ⌋ | E | E ∈F Define λ ( P ) = lim n →∞ λ n ( P ). We have e ( P ) ≤ π ( P ) ≤ λ ( P ) if both limits exist. There are posets that have π ( P ) < λ ( P ). Example: q q ❆✁ V 2 = 1 = π ( V 2 ) < λ ( V 2 ) = 2 . q

There are posets that have λ n ( P ) ≤ e ( P ) for all n . For such posets e ( P ) = π ( P ) = λ ( P ). Example: The chain poset of size k , P k is a poset that has such property.

There are posets that have λ n ( P ) ≤ e ( P ) for all n . For such posets e ( P ) = π ( P ) = λ ( P ). Example: The chain poset of size k , P k is a poset that has such property. We call such a poset a uniformly L-bounded poset.

There are posets that have λ n ( P ) ≤ e ( P ) for all n . For such posets e ( P ) = π ( P ) = λ ( P ). Example: The chain poset of size k , P k is a poset that has such property. We call such a poset a uniformly L-bounded poset. Various uniformly L-bounded posets: s s s s s s � ❅ � � ❏ � ✂ ❅ ✂ ❇ ✟✟ ❍ ❏ � ❇ ❅ s s s s s s s ❍ � ❅ s s s s s s ✡ ❅ ❇ � ✂ � s s s s s s s s s s s ❅ � � ❅ � ❅ ✡ ❅ � ❅ ❇ ✂ s s s s s s

Proposition (Griggs, Li, and Lu, 2011) For a uniformly L-bounded poset P with e ( P ) = m, La ( n , P ) = Σ( n , m ) for all n . If F is a P-free family of the largest size, then F = B ( n , m ) .

Theorem (Griggs, Li, and Lu, 2011) The k-diamond poset D k is a uniformly L-bounded poset if k is an � m integer in [2 m − 1 − 1 , 2 m − � − 1] for some integer m ≥ 2 . ⌊ m 2 ⌋ s ✚ ❩❩❩❩ � ❅ ✚ � ❅ ✚ ✚ � ❅ ............. D k s s s s ❩ ✚✚✚✚ ❅ � ❩ � �� � ❅ � k ❩ ❩ ❅ � s Example: k = 3 , 4 , 7 , 8 , 9 , 15 , 16 , ....

Theorem (Griggs, Li, and Lu, 2011) The harp poset H ( ℓ 1 , ..., ℓ k ) is a uniformly L-bounded poset if ℓ 1 > · · · > ℓ k ≥ 3 . ✘✘✘✘ ✏✏✏ s ✑✑ ✁ ❇ ❇ s s s ✁ ❇ s s s ❇ s s s ✂ H (7 , 6 , 5 , 4 , 3) s s s ❆ ✂ s ❳❳❳❳ PPP ◗◗ s s ❆ s ✂ ✂

Let Q be a subset of poset P . Then Q + := { p ∈ P | q ≤ p for some q ∈ Q } . Q − := { p ∈ P | p ≤ q for some q ∈ Q } . Lemma For a poset P, if there exists an element p ∈ P such that P = P 1 ∪ P 2 , where P 1 = { p } − and P 2 = { p } + , then e ( P ) ≥ e ( P 1 ) + e ( P 2 ) and λ n ( P ) ≤ λ n ( P 1 ) + λ n ( P 2 ) . s � ❏ ❏ s s P 2 s ✡ s ❅ ✡ s p s � ❅ ❍ ✟✟ s s P 1 ❍ s s ❅ � s

If a poset has the unique maximal(minimal) element, then the element is denoted by ˆ 1(ˆ 0).

If a poset has the unique maximal(minimal) element, then the element is denoted by ˆ 1(ˆ 0). Theorem Let P 1 , . . . , P k be uniformly L-bounded posets such that each of which has ˆ 0 and ˆ 1 . By identifying the ˆ 1 of P i to the ˆ 0 of P i +1 , we obtain a new uniformly L-bounded poset. s � ❅ s s s P k ❅ � . s . . s � ❅ s s s P 2 ❅ � s s � ❅ s s s P 1 ❅ � s