Largest families of sets under conditions defined by a given poset - PowerPoint PPT Presentation

Largest families of sets under conditions defined by a given poset Gyula O.H. Katona R´ enyi Institute, Budapest Jiao Tong University Combinatorics Seminar October 21, 2014

An ancient combinatorial problem Let [ n ] = { 1 , 2 , . . . , n } be a finite set. Question. Find the maximum number of subsets A of [ n ] such that A �⊂ B holds for them. 1

Theorem (Sperner, 1928) If A is a family of distinct subsets of [ n ] without inclusion ( A, B ∈ A implies A �⊂ B ) then � n � |A| ≤ . � n � 2 2

Theorem (Sperner, 1928) If A is a family of distinct subsets of [ n ] without inclusion ( A, B ∈ A implies A �⊂ B ) then � n � |A| ≤ . � n � 2 Sharp! 3

Theorem (Sperner, 1928) If A is a family of distinct subsets of [ n ] without inclusion ( A, B ∈ A implies A �⊂ B ) then � n � |A| ≤ . � n � 2 Sharp! Exact maximum! 4

Theorem (Erd˝ os, 1938) If A is a family of distinct subsets of [ n ] without k + 1 distinct members satisfying A 1 ⊂ A 2 ⊂ . . . ⊂ A k +1 ∈ A then � |A| ≤ k largest binomial coefficients of order n. 5

Theorem (Erd˝ os, 1938) If A is a family of distinct subsets of [ n ] without k + 1 distinct members satisfying A 1 ⊂ A 2 ⊂ . . . ⊂ A k +1 ∈ A then � |A| ≤ k largest binomial coefficients of order n. Sharp! Exact maximum! 6

Consider generalizations , where the excluded pattern is expressed only by inclusion . 7

Partially ordered set (Poset) P = ( X, < ) where at most one of <, = , > holds, (i) (ii) < is transitive. ( a, b ∈ X are comparable in P iff a < b, a = b or a > b .) In our case X = 2 [ n ] and A < B in this poset iff A ⊂ B . Notation: B n = (2 [ n ] , ⊂ ) . 8

9

Illustration of the Sperner theorem, poset form 10

Generalizations of Sperner theorem Notation. La ( n, P ) = the maximum number of elements of B n such that the poset induced by these elements does not contain P as a subposet . 11

12

Example 1: P = I Theorem (Sperner) � n � La ( n, I ) = . ⌊ n 2 ⌋ 13

Example 2: P = P k +1 Theorem (Erd˝ os) La ( n, P k +1 ) = � k largest binomial coefficients of order n. 14

Example 3: V r = { a, b 1 , . . . , b r } a < b 1 , . . . a < b r where 15

Construction for V 2 . 16

� n � Find many sets A 1 , . . . , A m of size + 1 2 � n � such that | A i ∩ A j | < ( i � = j ) . 2 This is an old open problem of coding theory. Known: � n � n � 1 � 1 � � 2 �� n ≤ max m ≤ n + O . � n � n � � n 2 2 2 Graham and Sloane, 1980 17

Theorem (K-Tarj´ an, 1983) � n � 1 � n � 1 � � �� � � �� 1 + 1 1 + 2 n + Ω ≤ La ( n, V 2 ) ≤ n + O . � n � n � � n 2 n 2 2 2 Hard to find the right constant. 18

Theorem (De Bonis-K, 2007) � n � 1 � n � 1 � � �� � � �� 1 + r 1 + 2 r n + Ω ≤ La ( n, V r +1 ) ≤ n + O . � n � n � � n 2 n 2 2 2 19

Theorem (Griggs-K, 2008) � n � 1 � n � 1 � � �� � � �� 1 + 1 1 + 2 n + Ω ≤ La ( n, N ) ≤ n + O . � n � n � � n 2 n 2 2 2 Remark. The estimates, up to the first two terms are the same as for V 2 . 20

But! It is interesting to mention that the “La” function will jump if the excluded The butterfly ⋊ poset contains one more relation. ⋉ contains 4 elements: a, b, c, d with a < c, a < d, b < c, b < d . Theorem (De Bonis, K, Swanepoel, 2005) Let n ≥ 3 . Then � � � � n n La( n, ⋊ ⋉ ) = + . ⌊ n/ 2 ⌋ ⌊ n/ 2 ⌋ + 1 21

But! It is interesting to mention that the “La” function will jump if the excluded poset contains one more relation. The butterfly ⋊ ⋉ contains 4 elements: a, b, c, d with a < c, a < d, b < c, b < d . Theorem (De Bonis, K, Swanepoel, 2005) Let n ≥ 3 . Then � � � � n n La( n, ⋊ ⋉ ) = + . ⌊ n/ 2 ⌋ ⌊ n/ 2 ⌋ + 1 A BOOK proof by P´ eter Burcsi and D´ aniel T. Nagy, 2013 . 22

The height h ( P ) of the poset P is the length of the longest chain . Theorem (Burcsi, D.T. Nagy, 2013) � | P | + h ( P ) � � n � La( n, P ) ≤ − 1 . 2 ⌊ n/ 2 ⌋ 23

The height h ( P ) of the poset P is the length of the longest chain . Theorem (Burcsi, D.T. Nagy, 2013) � | P | + h ( P ) � � n � La( n, P ) ≤ − 1 . 2 ⌊ n/ 2 ⌋ They also found infinitely many P with equality. 24

The height h ( P ) of the poset P is the length of the longest chain . Theorem (Burcsi, D.T. Nagy, 2013) � | P | + h ( P ) � � n � La( n, P ) ≤ − 1 . 2 ⌊ n/ 2 ⌋ They also found infinitely many P with equality. Improvements of the bound by Hong-Bin Chen and Wei-Tian Li . 25

asymmetric butterfly 26

asymmetric butterfly Theorem (Abishek Methuku and Casey Tompkins) Let n ≥ 3 . Then � n � � n � La( n, ) = + . ⌊ n/ 2 ⌋ ⌊ n/ 2 ⌋ + 1 27

Results for trees A poset is a tree if it Hasse diagram is a tree. 28

Results for trees A poset is a tree if it Hasse diagram is a tree. Theorem (Griggs-Linyuan Lincoln Lu, 2009) Let T be a tree and suppose that it has two levels, then � n � n � � � 1 �� � � � 1 �� 1 + Ω ≤ La( n, T ) ≤ 1 + O . ⌊ n ⌊ n 2 ⌋ n 2 ⌋ n Theorem (Bukh, 2010) Let T be a tree. Then � n � n � � � 1 �� � � � 1 �� ( h ( T ) − 1) 1 + Ω ≤ La( n, T ) ≤ ( h ( T ) − 1) 1 + O . ⌊ n ⌊ n 2 ⌋ n 2 ⌋ n 29

The diamond problem Let P be the following poset: P = D is called the diamond . � � � � n n + ≤ La( n, D ) . ⌊ n/ 2 ⌋ ⌊ n/ 2 ⌋ + 1 (Two middle levels.) 30

Theorem (Axenovich-Manske-Martin, 2012) � n � � � 1 �� La( n, D ) ≤ 2 . 283 1 + O . ⌊ n 2 ⌋ n 31

Theorem (Axenovich-Manske-Martin, 2012) � n � � � 1 �� La( n, D ) ≤ 2 . 283 1 + O . ⌊ n 2 ⌋ n Theorem (Griggs-Lu Linyuan-Li Wei-Tian, 2012) � n � � � 1 �� La( n, D ) ≤ 2 . 273 1 + O . ⌊ n 2 ⌋ n 32

Theorem (Axenovich-Manske-Martin, 2012) � n � � � 1 �� La( n, D ) ≤ 2 . 283 1 + O . ⌊ n 2 ⌋ n Theorem (Griggs-Lu Linyuan-Li Wei-Tian, 2012) � n � � � 1 �� La( n, D ) ≤ 2 . 273 1 + O . ⌊ n 2 ⌋ n Theorem (Kramer-Martin-Young, 2012) � n � � � 1 �� La( n, D ) ≤ 2 . 25 1 + O . ⌊ n 2 ⌋ n 33

An easier problem: restrict ourselves to the 3 middle levels. 34

An easier problem: restrict ourselves to the 3 middle levels. Theorem (Manske-Shen, 2013) If F is in the 3 middle levels and contains no diamond then � n � |F| ≤ (2 . 15471 + o (1)) . ⌊ n 2 ⌋ 35

An easier problem: restrict ourselves to the 3 middle levels. Theorem (Manske-Shen, 2013) If F is in the 3 middle levels and contains no diamond then � n � |F| ≤ (2 . 15471 + o (1)) . ⌊ n 2 ⌋ Theorem (Balogh-Hu-Lidick´ y-Liu, 2014) If F is in the 3 middle levels and contains no diamond then � n � |F| ≤ (2 . 15121 + o (1)) . ⌊ n 2 ⌋ 36

Czabarka, Dutle, Johnston and Sz´ ekely recently gave a class of very nice algebraic constructions with asymptotic equality. 37

Conjecture (everybody) La( n, P ) � n � ⌊ n 2 ⌋ tends to an integer for every poset P . 38

Conjecture (everybody) La( n, P ) � n � ⌊ n 2 ⌋ tends to an integer for every poset P . The first application of these type of theorems (namely Bukh’s theorem ) is due to Patk´ os . 39

A new type of generalization of the Sperner theorem. Given a ”small” poset P , find the maximum number of copies of P in B n in such a way that no two elements in different copies are comparable . 40

A new type of generalization of the Sperner theorem. Given a ”small” poset P , find the maximum number of copies of P in B n in such a way that no two elements in different copies are comparable . 41

A new type of generalization of the Sperner theorem. Given a ”small” poset P , find the maximum number of copies of P in B n in such a way that no two elements in different copies are comparable . 42

This maximum is denoted by LA ( n, P ) . The longest path in P is h ( P ) . Theorem � n − h ( P )+1 � LA( n, P ) ≤ . ⌊ n − h ( P )+1 ⌋ 2 This is sharp when P is a subposet of B h ( P ) − 1 : 43

44

Observation Suppose P ∗ is an embedding into B n , a < b < c ∈ B n , a, c ∈ P ∗ . If d is incomparable with the elements of P ∗ then b and d are also incomparable. 45

Observation Suppose P ∗ is an embedding into B n , a < b < c ∈ B n , a, c ∈ P ∗ . If d is incomparable with the elements of P ∗ then b and d are also incomparable. Proof Indirect way: 46

Observation Suppose P ∗ is an embedding into B n , a < b < c ∈ B n , a, c ∈ P ∗ . If d is incomparable with the elements of P ∗ then b and d are also incomparable. Proof Indirect way: 47

Convex closure of P ∗ : add all such b s cc ( P ∗ ) = P ∗ ∪ { b : a < b < c for some a, c ∈ P ∗ } . 48