Packing Posets in the Boolean Lattice

Transcription

1 page.1... Packing Posets in the Boolean Lattice Andrew P. Dove Jerrold R. Griggs University of South Carolina Columbia, SC USA SIAM-DM14 Conference Minneapolis

2 page.2 Andrew Dove

3 page.3 Andrew Dove Jerry Griggs

4 page.4 For a poset P, we consider how large a family F of subsets of [n] := {1,..., n} we may have in the Boolean Lattice B n : (2 [n], ) containing no (weak) subposet P. We are interested in determining or estimating La(n, P) := max{ F : F 2 [n], P F}.

5 page.5 For a poset P, we consider how large a family F of subsets of [n] := {1,..., n} we may have in the Boolean Lattice B n : (2 [n], ) containing no (weak) subposet P. We are interested in determining or estimating La(n, P) := max{ F : F 2 [n], P F}. Example For the poset P = N, F means F contains no 4 subsets A, B, C, D such that A B, C B, C D. Note that A C is allowed: The subposet does not have to be induced.

6 page.6 The Boolean Lattice B 4 {1, 2, 3, 4} {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} {1} {2} {3} {4}

7 page.7 A Family of Subsets F in B 4 {1, 2, 3, 4} 2 2 {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} {1} {2} {3} {4} 2

8 page.8 F contains the poset N {1, 2, 3, 4} 2 2 {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} {1} {2} {3} {4} 2

9 page.9 A Large N -free Family in B 4 {1, 2, 3, 4} 2 {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} 2 {1} {2} {3} {4}

10 page.10 Given a finite poset P, we are interested in determining or estimating La(n, P) := max{ F : F 2 [n], P F}.

11 page.11 Given a finite poset P, we are interested in determining or estimating La(n, P) := max{ F : F 2 [n], P F}. For many posets, La(n, P) is exactly equal to the sum of middle k binomial coefficients, denoted by Σ(n, k). Moreover, the largest families may be B(n, k), the families of subsets of middle k sizes. [n] } k levels

12 page.12 Foundational results: Let P k denote the k-element chain (path poset). Theorem (Sperner, 1928) For all n, La(n, P 2 ) = and the extremal families are B(n, 1). ( ) n n 2,

13 page.13 Foundational results: Let P k denote the k-element chain (path poset). Theorem (Sperner, 1928) For all n, La(n, P 2 ) = and the extremal families are B(n, 1). ( ) n n 2, Theorem (Erdős, 1945) For general k and n, La(n, P k ) = Σ(n, k 1), and the extremal families are B(n, k 1).

14 page.14 Excluded subposet P La(n, P) P 2 ( n n 2 ) [Sperner, 1928] Path P k, k 2. Σ(n, k 1) (k 1) ( ) n [P. Erdős, 1945] n 2 r-fork V r r {}}{... ( n n 2 ) [Katona-Tarján, 1981] [DeBonis-Katona 2007]

15 page.15 Excluded subposet P La(n, P) Butterfly B N K r,s(r, s 2) r {... }}{... }{{} s r, s 2 Σ(n, 2) 2 ( ) n n 2 ( n n 2 ) 2 ( n n 2 ) [DeBonis-Katona- Swanepoel, 2005] [G.-Katona, 2008] [De Bonis-Katona, 2007]

16 page.16 Asymptotic behavior of La(n, P) Definition La(n,P) π(p):= lim n ( n ). 2

17 page.17 Asymptotic behavior of La(n, P) Definition La(n,P) π(p):= lim n ( n ). 2 Conjecture (G.-Lu, 2008) For all P, π(p) exists and is integer. When Saks and Winkler (2008) observed what π(p) is in known cases, it led to the stronger Conjecture (G.-Lu, 2009) For all P, π(p) = e(p), where Definition e(p):= max m such that for all n, P B(n, m).

18 page.18 On the Diamond D 2 Problem Despite considerable effort it remains open to determine the value π(d 2 ) or even to show it exists! The conjectured value of π(d 2 ) is its lower bound, e(d 2 ) = 2.

19 page.19 Successive upper bounds on π(d 2 ): 2.5 [G.-Li, 2007] [G.-Li-Lu, 2008] [Axenovich-Manske-Martin, 2011] [G.-Li-Lu, 2011] 2.25 [Kramer-Martin-Young, 2012]

20 page.20 Three level problem To make things simpler, what if we restrict attention to D 2 -free families in the middle three levels of the Boolean lattice B n? We should get better upper bounds on F / ( n n 2 ) : [Axenovich-Manske-Martin, 2011] [Manske-Shen, 2012] [Balogh-Hu-Lidický-Liu, 2012]

21 page.21 Excluding a Family of Posets Let us now generalize La(n, P). We consider La(n, {P i }) := max{ F : F 2 [n], ip i F}. In words, it is the maximum size of a family F B n that contains no copy of any poset P i {P i }.

22 page.22 Starting Point What is La(n, {V, Λ}), where V = V 2?

23 page.23 Starting Point What is La(n, {V, Λ}), where V = V 2? Theorem (Katona, ( Tarján ) (1983)) n 1 La(n, {V, Λ}) = 2 n 1 2 ( ) n n. 2

24 page.24 Starting Point What is La(n, {V, Λ}), where V = V 2? Theorem (Katona, ( Tarján ) (1983)) n 1 La(n, {V, Λ}) = 2 n 1 2 ( ) n n. 2 Although solved, let us think about this question. A family with neither V nor Λ is constructed from subposets B 0 and B 1 that are not only disjoint, but are unrelated. It means that no element of one is related to an element of another.

25 page.25 Starting Point What is La(n, {V, Λ}), where V = V 2? Theorem (Katona, ( Tarján ) (1983)) n 1 La(n, {V, Λ}) = 2 n 1 2 ( ) n n. 2 Although solved, let us think about this question. A family with neither V nor Λ is constructed from subposets B 0 and B 1 that are not only disjoint, but are unrelated. It means that no element of one is related to an element of another. Our question becomes: What is the largest size of a family in B n constructed from pairwise unrelated copies of B 0 and B 1?

26 page.26 Maximum Packings We consider Pa(n, {P i }) := max{ F : F 2 [n] }, where each component of F is a copy of some poset in the collection {P i }. The collection may be infinite.

27 page.27 Maximum Packings We consider Pa(n, {P i }) := max{ F : F 2 [n] }, where each component of F is a copy of some poset in the collection {P i }. The collection may be infinite. This can be viewed as a generalization of the La(n, {Q j }) problem: La(n, {Q j }) = Pa(n, {P i }), where {P i } consists of all connected posets that do not contain a copy of any Q j {Q j }.

28 page.28 Maximum Packings We consider Pa(n, {P i }) := max{ F : F 2 [n] }, where each component of F is a copy of some poset in the collection {P i }. The collection may be infinite. This can be viewed as a generalization of the La(n, {Q j }) problem: La(n, {Q j }) = Pa(n, {P i }), where {P i } consists of all connected posets that do not contain a copy of any Q j {Q j }. We have that La(n, {V, Λ}) = Pa(n, {B 0, B 1 }).

29 page.29 Old Examples with New Notation Problem What is Pa(n, P) for a general poset P?

30 page.30 Old Examples with New Notation Problem What is Pa(n, P) for a general poset P? Note: Katona independently introduced what is equivalent to Pa(n, P) in Theorem (Sperner (1928)) Pa(n, B 0 ) is ( n n 2 ).

31 page.31 Old Examples with New Notation Problem What is Pa(n, P) for a general poset P? Note: Katona independently introduced what is equivalent to Pa(n, P) in Theorem (Sperner (1928)) Pa(n, B 0 ) is ( n n 2 ). Theorem (G., Stahl, Trotter (1984)) For the chain (or ( path) P k ) on k 1 elements, n k + 1 Pa(n, P k ) = k n k+1. 2 The maximum number of unrelated copies of P k in B n is 1 ( asymptotic to n ) 2 k 1 n. 2

32 page.32 Example: Packing many copies of V n+1 2 A1 A2 B123 B124 C12345 C A 1, 2 / A B12 1, 2, 3, 4 / B C1234 1, 2, 3, 4, 5, 6 / C

33 page.33 Example: Packing many copies of V n+1 2 A1 A2 B123 B124 C12345 C A 1, 2 / A B12 1, 2, 3, 4 / B C1234 1, 2, 3, 4, 5, 6 / C Pa(n, V)/ V [ (n ) ( ) ( ) ] 2 n 4 n 6 n 1 + n 1 + n

34 page.34 Example: Packing many copies of V n+1 2 A1 A2 B123 B124 C12345 C A 1, 2 / A B12 1, 2, 3, 4 / B C1234 1, 2, 3, 4, 5, 6 / C Pa(n, V)/ V [ (n ) ( ) ( ) ] 2 n 4 n 6 n 1 + n 1 + n [ ( ) 1 n 2 2 n ( ) n 2 4 n ( ) ] n 2 6 n = 1 ( ) n 3 n 2.

35 page.35 Example: Packing many copies of V On the other hand, by counting how many maximal chains in B n hit a V, we get the same expression as an asymptotic upper bound, and deduce Pa(n, V)/ V 1 3 ( ) n n 2.

36 page.36 Example: Packing many copies of V On the other hand, by counting how many maximal chains in B n hit a V, we get the same expression as an asymptotic upper bound, and deduce Pa(n, V)/ V 1 3 ( ) n n 2. Discussing this with Richard Anstee, we were led to make the Conjecture For every poset P, there is some integer c(p) such that Pa(n, P)/ P 1 ( n ) c(p) n as n. 2

37 page.37 Convex Closure and Main Theorem For a family F B n, its convex closure is the family F := {S B n A S B for some A, B F}. Notice that if a family G is unrelated to a family F, then G is also unrelated to F.

38 page.38 Convex Closure and Main Theorem For a family F B n, its convex closure is the family F := {S B n A S B for some A, B F}. Notice that if a family G is unrelated to a family F, then G is also unrelated to F. A family F B n is called convex if F = F.

39 page.39 Convex Closure and Main Theorem For a family F B n, its convex closure is the family F := {S B n A S B for some A, B F}. Notice that if a family G is unrelated to a family F, then G is also unrelated to F. A family F B n is called convex if F = F. We define c(p) to be the smallest size of a convex family of subsets containing a copy of P.

40 page.40 Convex Closure and Main Theorem For a family F B n, its convex closure is the family F := {S B n A S B for some A, B F}. Notice that if a family G is unrelated to a family F, then G is also unrelated to F. A family F B n is called convex if F = F. We define c(p) to be the smallest size of a convex family of subsets containing a copy of P. Theorem (D., G. (2013), and Katona, Nagy (2013)) As n goes to infinity, Pa(n, P) P ( n ) c(p) n. 2

41 page.41 Proof Sketch: Upper Bound of Pa(n, P) Theorem (D., G. (2013), and Katona, Nagy (2013)) As n goes to infinity, Pa(n, P) P ( n ) c(p) n. 2 No full (maximum) chain in B n meets more than one closure of a copy of P.

42 page.42 Proof Sketch: Upper Bound of Pa(n, P) Theorem (D., G. (2013), and Katona, Nagy (2013)) As n goes to infinity, Pa(n, P) P ( n ) c(p) n. 2 No full (maximum) chain in B n meets more than one closure of a copy of P. If each closure of a copy of P meets at least x full chains, then Pa(n, P) x n!; P Pa(n, P) P n! x.

43 page.43 Proof Sketch: Upper Bound of Pa(n, P) Theorem (D., G. (2013), and Katona, Nagy (2013)) As n goes to infinity, Pa(n, P) P ( n ) c(p) n. 2 No full (maximum) chain in B n meets more than one closure of a copy of P. If each closure of a copy of P meets at least x full chains, then Pa(n, P) x n!; P Pa(n, P) P n! x. We need to show the closure of a copy of P meets at least c(p) n/2! n/2! full chains asymptotically.

44 page.44 Upper Bound of Pa(n, P) Let a(n, m) denote the largest integer such that any family F B n, F = m, meets at least a(n, m) full chains.

45 page.45 Upper Bound of Pa(n, P) Let a(n, m) denote the largest integer such that any family F B n, F = m, meets at least a(n, m) full chains. Lemma As n goes to infinity, a(n, m) m n/2! n/2!.

46 page.46 Upper Bound of Pa(n, P) Let a(n, m) denote the largest integer such that any family F B n, F = m, meets at least a(n, m) full chains. Lemma As n goes to infinity, a(n, m) m n/2! n/2!. Proof of Lemma: Inclusion/Exclusion: Let F B n be a family of size m that meets a(n, m) full chains. Let b({a 1,..., A k }) be the number of full chains in B n that meet all of the sets in {A 1,..., A k }. a(n, m) A F b({a}) A 1,A 2 F b({a 1, A 2 }).

47 page.47 Upper Bound of Pa(n, P) Proof continued: a(n, m) b({a}) A F b({a 1, A 2 }) A 1,A 2 F

48 page.48 Upper Bound of Pa(n, P) Proof continued: a(n, m) b({a}) b({a 1, A 2 }) A F A 1,A 2 F ( b({a}) ) b({a, A 1 }) A F A 1 F n A n 2 2 A 1

49 page.49 Upper Bound of Pa(n, P) Proof continued: a(n, m) b({a}) b({a 1, A 2 }) A F A 1,A 2 F ( b({a}) ) b({a, A 1 }) A F A 1 F n A n 2 2 A 1 = ( b({a}) 1 ) b({a, A 1 }) b({a}) A F A 1 F n A n 2 2 A 1

50 page.50 Upper Bound of Pa(n, P) Proof continued: A F b({a}) a(n, m) b({a}) b({a 1, A 2 }) A F A 1,A 2 F ( b({a}) ) b({a, A 1 }) A F A 1 F n A n 2 2 A 1 = ( b({a}) 1 ) b({a, A 1 }) b({a}) A F A 1 F n A n 2 2 A 1 ( ) 1 2m n A F b({a}) m (n/2)! (n/2)!.

51 page.51 Lower Bound of Pa(n, P) By an elaborate extension of what we saw for P = V, we construct an F n B n, of unrelated copies of P, where as n, ( (2 k c(p)) j )( ) n F n P (2 k ) j+1 n j=0 2 = P 1 [ ]( ) 1 n 2 k 1 2k c(p) n 2 k 2 which gives the Theorem. = P c(p) ( ) n n 2,

52 page.52 Packing Induced Copies of P Denote by Pa (n, {P i }) the maximum size of a family F B n, where each connected component is an induced copy of a poset from the collection {P i }.

53 page.53 Packing Induced Copies of P Denote by Pa (n, {P i }) the maximum size of a family F B n, where each connected component is an induced copy of a poset from the collection {P i }. Let c (P) be the smallest size of a convex family containing an induced copy of P.

54 page.54 Packing Induced Copies of P Denote by Pa (n, {P i }) the maximum size of a family F B n, where each connected component is an induced copy of a poset from the collection {P i }. Let c (P) be the smallest size of a convex family containing an induced copy of P. Theorem (D., G. (2013) and Katona, Nagy (2013)) As n, Pa ( (n, P) n ) n. 2 P c (P)

55 page.55 Other Results For a finite collection of posets: Theorem (D., G. (2013)) As n goes to infinity, ( ) Pa(n, {P 1, P 2,..., P k }) max Pi ( n ) 1 i k c(p i ) n. 2 Theorem (D., G. (2013)) As n goes to infinity, ( ) Pa (n, {P 1, P 2,..., P k }) max Pi ( n ) 1 i k c (P i ) n. 2

56 page.56 Future Work Finding La(n, {P i }), even asymptotically.

57 page.57 Future Work Finding La(n, {P i }), even asymptotically. Finding Pa(n, {P i }) asymptotically for an infinite collection of posets.

58 page.58 Future Work Finding La(n, {P i }), even asymptotically. Finding Pa(n, {P i }) asymptotically for an infinite collection of posets. Finding exact values of Pa(n, P).

59 page.59 Future Work Finding La(n, {P i }), even asymptotically. Finding Pa(n, {P i }) asymptotically for an infinite collection of posets. Finding exact values of Pa(n, P). Designing an algorithm that quickly finds c(p), or even the complexity of such an algorithm.

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