Building Diamond-free Posets

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1 Aaron AMS Southeastern Sectional October 5, 2013 Joint with Éva Czabarka, Travis Johnston, and László Székely

2 The Diamond Conjecture is False Aaron AMS Southeastern Sectional October 5, 2013 Joint with Éva Czabarka, Travis Johnston, and László Székely The Diamond Conjecture is False

4 Aaron AMS Southeastern Sectional October 5, 2013 Joint with Éva Czabarka, Travis Johnston, and László Székely

6 Two Interesting Posets In B 8 : Take the size 5 level, the size 3 level

7 Two Interesting Posets In B 8 : Take the size 5 level, the size 3 level and an S(3, 4, 8) Steiner system on the size 4 level.

8 Two Interesting Posets In B 8 : Take the size 5 level, the size 3 level and an S(3, 4, 8) Steiner system on the size 4 level. No Diamonds! (Steiner)

9 Two Interesting Posets In B 8 : Take the size 5 level, the size 3 level and an S(3, 4, 8) Steiner system on the size 4 level. No Diamonds! (Steiner) Same number of sets as the two largest levels.

10 Two Interesting Posets In B 12 : Take the size 7 level, the size 5 level and an S(5, 6, 12) Steiner system on the size 6 level. No Diamonds! (Steiner) Same number of sets as the two largest levels.

11 Two Interesting Posets This works for any B 2d, if there s an S(d 1, d, 2d). But no Steiner system is know with t > 5.

12 Two Interesting Posets This works for any B 2d, if there s an S(d 1, d, 2d). But no Steiner system is know with t > 5. Plus,

13 Cayley Posets Abelian Group Γ, Generating set H, and its Digraph D.

14 Cayley Posets Abelian Group Γ, Generating set H, and its Digraph D. Copy the group, edges in D become covers.

15 Cayley Posets Abelian Group Γ, Generating set H, and its Digraph D. Copy the group, edges in D become covers. Repeat in both directions. Any finite subposet of this is a Cayley Poset.

16 Some Definitions A Cayley Poset... is aperiodic if gcd{k h 1 + h h k = 0, h i H} = 1.

17 Some Definitions A Cayley Poset... is aperiodic if gcd{k h 1 + h h k = 0, h i H} = 1. has a strong chain if there are (g 1, i 1 ) < (g 2, i 2 ) < (g 3, i 3 ), where g 2 g 1 = i 2 i 1 j=1 h j and g 3 g 2 = i 3 i 2 j=1 h j can be written sharing a common term.

18 Some Definitions A Cayley Poset... is aperiodic if gcd{k h 1 + h h k = 0, h i H} = 1. has a strong chain if there are (g 1, i 1 ) < (g 2, i 2 ) < (g 3, i 3 ), where g 2 g 1 = i 2 i 1 j=1 h j and g 3 g 2 = i 3 i 2 j=1 h j can be written sharing a common term. is strongly diamond-free if it has no diamonds, and no strong chains.

19 Back to B n Assign an element from H to each element of [n]. w : [n] H. Extend linearly to subsets A [n].

20 Back to B n Assign an element from H to each element of [n]. w : [n] H. Extend linearly to subsets A [n]. For an element g Γ and i N, S g (i) = {A : w(a) = g and A = n/2 + i}.

21 Back to B n If w is a weighting, and P = {(g j, i j )} is a Cayley poset with l elements, there s a family of sets F(n, w, P) = l S gj (i j ) j=1

22 Back to B n If w is a weighting, and P = {(g j, i j )} is a Cayley poset with l elements, there s a family of sets F(n, w, P) = l S gj (i j ) j=1 Lemma If P is strongly diamond free, then F(n, w, P) is diamond free.

23 Back to B n If w is a weighting, and P = {(g j, i j )} is a Cayley poset with l elements, there s a family of sets F(n, w, P) = l S gj (i j ) j=1 Lemma If P is strongly diamond free, then F(n, w, P) is diamond free. If we can find good weights, and good posets, we have interesting examples!

24 Random is Good Instead of assigning weights, let them be random variables. { 1/ H if g H P[w(x) = g] =. 0 o/w

25 Random is Good Instead of assigning weights, let them be random variables. { 1/ H if g H P[w(x) = g] =. 0 o/w For a A [n], consider finding w(a) as a Markov process, adding the weight of each element to a running total.

26 An Old Theorem Theorem For any group G, and H G, if H generates G, and is not contained in some coset of a normal subgroup, the (infinite) Markov process described is irreducible, aperiodic, and converges to the uniform distribution on G.

27 An Old Theorem Theorem For any group G, and H G, if H generates G, and is not contained in some coset of a normal subgroup, the (infinite) Markov process described is irreducible, aperiodic, and converges to the uniform distribution on G. Aperiodic Cayley posets have these properties! Hence for n large, i fixed, for all g and A = n/2 + i 1/ G ɛ P[w(A) = g] 1/ G + ɛ

28 An Old Theorem Theorem For any group G, and H G, if H generates G, and is not contained in some coset of a normal subgroup, the (infinite) Markov process described is irreducible, aperiodic, and converges to the uniform distribution on G. Aperiodic Cayley posets have these properties! Hence for n large, i fixed, for all g and A = n/2 + i 1/ G ɛ E ( S g(i) ) ) 1/ G + ɛ ( n n/2 +i

29 A New Result Note that if P is strongly diamond free, l j=1 S g j (i j ) = F(n, w, P) La(n, D 2 ).

30 A New Result Note that if P is strongly diamond free, l j=1 S g j (i j ) = F(n, w, P) La(n, D 2 ). Theorem If G is an abelian group with G = m and an aperiodic generating set H, and P is a strongly diamond-free Cayley poset with l elements, l m = lim n E [ F(n, w, P) ] ) lim inf ( n n/2 n La(n, D 2 ) ). ( n n/2

31 A New Result Note that if P is strongly diamond free, l j=1 S g j (i j ) = F(n, w, P) La(n, D 2 ). Theorem If G is an abelian group with G = m and an aperiodic generating set H, and P is a strongly diamond-free Cayley poset with l elements, l m = lim n E [ F(n, w, P) ] ) lim inf ( n n/2 n La(n, D 2 ) ). ( n n/2 A single Cayley poset with l > 2m would refute the diamond conjecture...

32 No Counterexamples Back in the infinite poset,

33 No Counterexamples Back in the infinite poset, Keep one generator.

34 No Counterexamples Back in the infinite poset, Keep one generator. Avoiding strong chains forces l 2m.

35 But Some Examples Example 1: l = 2m Any group, any generating set, and two levels (this is not interesting)...

36 But Some Examples Example 2: l = 2m Z m, H is two elements with gcd(a, b) = 1. (g, 3) : g a + b (a, 2), (b, 2) (g, 1) : g 0 (sometimes periodic...)

37 But Some Examples Example 3: l = 2m 1 Z 7, generators H = {2, 3, 5} (g, 3) : g 0, 1, 5 (2, 2), (3, 2), (5, 2) (g, 1) : g 0

38 But Some Examples Example 4: l = 2m 2 Z 4k+1, generators H = {2k, 2k + 1} (g, 4) : g = k + 2, k + 3,... 3k 2, (g, i) : g = k, k + 1,..., 3k, and i = 1, 2, 3 This is on four levels, and can be made as tight to the conjecture as desired.

39 The Last Slide Can this be used for other forbidden posets? Find conditions on the Cayley poset for what you want.

40 The Last Slide Can this be used for other forbidden posets? Find conditions on the Cayley poset for what you want. Can we show that there are always distant outliers in the expectation? (or any?)

41 The Last Slide Can this be used for other forbidden posets? Find conditions on the Cayley poset for what you want. Can we show that there are always distant outliers in the expectation? (or any?) Is there a single diamond-free poset with more than the largest two levels worth of elements?

42 Thanks! For the full paper, go to the arxiv:

Packing posets in a family of subsets

Packing posets in a family of subsets Andrew P. Dove Jerrold R. Griggs University of South Carolina Columbia, SC 29208 October 5, 2013 B n is the inclusion poset of all subsets of {1, 2,..., n}. Definition (La(n, P)) The quantity La(n, P)