Packing posets in a family of subsets

Transcription

1 Andrew P. Dove Jerrold R. Griggs University of South Carolina Columbia, SC October 5, 2013

2 B n is the inclusion poset of all subsets of {1, 2,..., n}. Definition (La(n, P)) The quantity La(n, P) is the maximum size of a family F B n that contains no copy of poset P.

3 B n is the inclusion poset of all subsets of {1, 2,..., n}. Definition (La(n, P)) The quantity La(n, P) is the maximum size of a family F B n that contains no copy of poset P. This may be generalized: Definition (La(n, {P i })) The quantity La(n, {P i }) is the maximum size of a family F B n that contains no copy of any poset P i {P i }.

4 Starting Point What is La(n, {V, Λ})?

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

6 Starting Point What is La(n, {V, Λ})? Theorem (Katona, Tarján (1983)) ( ) n 1 La(n, {V, Λ}) = 2 n 1 2. A family with neither a V nor a Λ is a family constructed from pairwise unrelated copies of B 0 and B 1.

7 Starting Point What is La(n, {V, Λ})? Theorem (Katona, Tarján (1983)) ( ) n 1 La(n, {V, Λ}) = 2 n 1 2. A family with neither a V nor a Λ is a family constructed from pairwise unrelated copies of B 0 and B 1. So same question: what is the size of the largest family in B n that is constructed from pairwise unrelated copies of B 0 and B 1?

8 Definition (Pa(n, {P i })) The quantity Pa(n, {P i }) is the maximum size of a family F B n, where each connected component is a copy of a poset from the collection {P i }.

9 Definition (Pa(n, {P i })) The quantity Pa(n, {P i }) is the maximum size of a family F B n, where each connected component is a copy of a poset from the collection {P i }. La(n, {Q j }) = Pa(n, {P i }), where {P i } is all connected posets which do not contain a copy of any Q j {Q j }. The Pa(n, {P i }) problem is a generalization of the La(n, {Q j }) problem.

10 Definition (Pa(n, {P i })) The quantity Pa(n, {P i }) is the maximum size of a family F B n, where each connected component is a copy of a poset from the collection {P i }. La(n, {Q j }) = Pa(n, {P i }), where {P i } is all connected posets which do not contain a copy of any Q j {Q j }. The Pa(n, {P i }) problem is a generalization of the La(n, {Q j }) problem. Now La(n, {V, Λ}) = Pa(n, {B 0, B 1 }).

11 Old Examples with New Notation What is Pa(n, P) for a general poset P?

12 Old Examples with New Notation What is Pa(n, P) for a general poset P? Theorem (Sperner (1928)) ( ) n Pa(n, B 0 ) is n 2.

13 Old Examples with New Notation What is Pa(n, P) for a general poset P? Theorem (Sperner (1928)) ( ) n Pa(n, B 0 ) is n 2. Theorem (Griggs, Stahl, Trotter (1984)) Where P k is the chain ( (or path) ) on k + 1 elements, n k Pa(n, P k ) is (k + 1) n k 2 k + 1 ( ) n 2 k n 2.

14 Old Examples with New Notation What is Pa(n, P) for a general poset P? Theorem (Sperner (1928)) ( ) n Pa(n, B 0 ) is n 2. Theorem (Griggs, Stahl, Trotter (1984)) Where P k is the chain ( (or path) ) on k + 1 elements, n k Pa(n, P k ) is (k + 1) n k 2 k + 1 ( ) n 2 k n 2. As n goes to infinity, maybe Pa(n, P) is asymptotically a constant multiple of ( ) n n 2.

15 Convex Closure Definition (convex closure) For a family F B n, F := {S B n A S B for some A, B F}.

16 Convex Closure Definition (convex closure) For a family F B n, F := {S B n A S B for some A, B F}. Definition (convex family) A family F B n is called convex if F = F.

17 Convex Closure Definition (convex closure) For a family F B n, F := {S B n A S B for some A, B F}. Definition (convex family) A family F B n is called convex if F = F. Definition (c(p)) The value c(p) is the size of the smallest convex family containing a copy of P.

18 Convex Closure Definition (convex closure) For a family F B n, F := {S B n A S B for some A, B F}. Definition (convex family) A family F B n is called convex if F = F. Definition (c(p)) The value c(p) is the size of the smallest convex family containing a copy of P. Theorem (D., G. (2013), and independently Katona, Nagy (2013)) ( ) As n goes to infinity, Pa(n, P) P n c(p) n 2.

19 Upper Bound of Pa(n, P) P c(p) n n 2 No full chain in B n may meet more than one closure of a copy of P.

20 Upper Bound of Pa(n, P) P c(p) n n 2 No full chain in B n may meet 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.

21 Upper Bound of Pa(n, P) P c(p) n n 2 No full chain in B n may meet 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. Need to show the closure of a copy of P meets at least c(p) n/2! n/2! full chains asymptotically.

22 Upper Bound of Pa(n, P) P c(p) n n 2 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.

23 Upper Bound of Pa(n, P) P c(p) n n 2 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!.

24 Upper Bound of Pa(n, P) P c(p) n n 2 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 }).

25 Upper Bound of Pa(n, P) P c(p) n n 2 Proof continued: a(n, m) b({a}) A F b({a 1, A 2 }) A 1,A 2 F

26 Upper Bound of Pa(n, P) P c(p) n n 2 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

27 Upper Bound of Pa(n, P) P c(p) n n 2 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

28 Upper Bound of Pa(n, P) P c(p) n n 2 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 F b({a}) ( 1 2m n A 1 F n A n 2 2 A 1 ) A F b({a}) m n/2! n/2!.

29 Lower Bound of Pa(n, P) P c(p) ( ) Example: Pa(n, V) V n c(v) n 2 = ( n n 2 n n 2 ) via construction.

30 Lower Bound of Pa(n, P) P c(p) ( ) Example: Pa(n, V) V n c(v) n 2 = ( n n 2 n n 2 ) via construction. 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

31 Lower Bound of Pa(n, P) P c(p) ( ) Example: Pa(n, V) V n c(v) n 2 = ( n n 2 n n 2 ) via construction. n+1 2 A1 A2 B123 B124 C12345 C Pa(n, V) V A 1, 2 / A B12 1, 2, 3, 4 / B C1234 1, 2, 3, 4, 5, 6 / C [ ( ) ( ) ( ) ] n 2 n 4 n 6 n n n

32 Lower Bound of Pa(n, P) P c(p) ( ) Example: Pa(n, V) V n c(v) n 2 = ( n n 2 n n 2 ) via construction. n+1 2 A1 A2 B123 B124 C12345 C Pa(n, V) V V A 1, 2 / A B12 1, 2, 3, 4 / B C1234 1, 2, 3, 4, 5, 6 / C [ ( ) ( ) ( ) ] n 2 n 4 n 6 n n n [ ( ) 1 n 2 2 n ( ) n 2 4 n ( ) ] n 2 6 n = V ( ) [ ] n 1 4 n

33 Lower Bound of Pa(n, P) P c(p) n n 2 In general, we construct an F n B n, where as n goes to infinity, ( (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 = P c(p) ( ) n n 2.

34 Other Results Definition (Pa (n, {P i })) The quantity Pa (n, {P i }) is 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 }.

35 Other Results Definition (Pa (n, {P i })) The quantity Pa (n, {P i }) is 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 }. Definition (c (P)) The value c (P) is the size of the smallest convex family containing an induced copy of P.

36 Other Results Definition (Pa (n, {P i })) The quantity Pa (n, {P i }) is 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 }. Definition (c (P)) The value c (P) is the size of the smallest convex family containing an induced copy of P. Theorem (D., G. (2013), and independently Katona, Nagy (2013)) ( ) As n goes to infinity, Pa (n, P) P n c (P) n 2.

37 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.

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

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

40 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).

41 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). Finding an algorithm that quickly finds c(p), or even the complexity of such an algorithm.