On families of subsets with a forbidden subposet
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1 On families of subsets with a forbidden subposet Linyuan Lu lu@math.sc.edu. University of South Carolina Joined work with Jerry Griggs On families of subsets with a forbidden subposet p./
2 Poset A poset is a set S together with a partial ordering on it. Reflexive Anti-symmetric a a a b and b a implies a = b Transitive a b and b c implies a c On families of subsets with a forbidden subposet p./
3 Posets and Hasse diagrams Chains P n = ([n], ) P P 3 P 4 On families of subsets with a forbidden subposet p.3/
4 Posets and Hasse diagrams Chains P n = ([n], ) P P 3 P 4 Boolean lattice Q n = ( [n], ) Q Q Q 3 On families of subsets with a forbidden subposet p.3/
5 Sperner s theorem Theorem(Sperner 98) Let F beafamily of subsetsof [n] = {,,...,n}. Suppose Forany A,B F,neither A B nor B A. Then ( ) n F n. On families of subsets with a forbidden subposet p.4/
6 Sperner s theorem Theorem(Sperner 98) Let F beafamily of subsetsof [n] = {,,...,n}. Suppose Forany A,B F,neither A B nor B A. Then ( ) n F n. YBLM inequality: Under the same condition, we have F F ( n ). F Yamomoto Bollobas Lubell Meshalkin On families of subsets with a forbidden subposet p.4/
7 Subposets A poset (S, ) contains a subposet (S, ) if there exists an injection f : S S which preserves partial orderings. f(a ) f(b ) if a b On families of subsets with a forbidden subposet p.5/
8 Subposets A poset (S, ) contains a subposet (S, ) if there exists an injection f : S S which preserves partial orderings. Subposets of Q 3 : f(a ) f(b ) if a b C 6 C 6 N 5 Q 3 On families of subsets with a forbidden subposet p.5/
9 Subposets A poset (S, ) contains a subposet (S, ) if there exists an injection f : S S which preserves partial orderings. Subposets of Q 3 : f(a ) f(b ) if a b C 6 C 6 N 5 Q 3 We say (S, ) is (S, )-free if no such injection f exists. On families of subsets with a forbidden subposet p.5/
10 La(n, H) For a fixed poset H, let La(n,H) denote the largest size of H-free family of subsets of [n]. On families of subsets with a forbidden subposet p.6/
11 La(n, H) For a fixed poset H, let La(n,H) denote the largest size of H-free family of subsets of [n]. Sperner(98): La(n,P ) = ( ) n n. On families of subsets with a forbidden subposet p.6/
12 La(n, H) For a fixed poset H, let La(n,H) denote the largest size of H-free family of subsets of [n]. Sperner(98): Erdős(938): La(n,P r ) = La(n,P ) = n+r i= n r+ ( ) n n. ( ) n i ( ) n = (r + o n ()) n. On families of subsets with a forbidden subposet p.6/
13 Trivial bounds of La(n, H) If H H, then La(n,H ) La(n,H). On families of subsets with a forbidden subposet p.7/
14 Trivial bounds of La(n, H) If H H, then La(n,H ) La(n,H). Every poset H on [n] is a subposet of chain P n. On families of subsets with a forbidden subposet p.7/
15 Trivial bounds of La(n, H) If H H, then La(n,H ) La(n,H). Every poset H on [n] is a subposet of chain P n. d(h) o n () La(n,H) ( n n ) H. Here depth d(h) is the order of the longest chain in H. On families of subsets with a forbidden subposet p.7/
16 Known results P Sperner(98): La(n,P ) = ( ) n n. On families of subsets with a forbidden subposet p.8/
17 Known results P Sperner(98): La(n,P ) = ( ) n n. P r Erdős(938): ( ) n La(n,P r ) = (r + o n ()) n. On families of subsets with a forbidden subposet p.8/
18 Known results P Sperner(98): La(n,P ) = ( ) n n. P r Erdős(938): ( ) n La(n,P r ) = (r + o n ()) n. V Tarján(975) La(n,V ) = ( + O( ( ) n n )) n On families of subsets with a forbidden subposet p.8/
19 Known asymptotic results V r Thanh(998)/DeBonis, Katona,(006) La(n,V r ) = ( + O( ( ) n n )) n On families of subsets with a forbidden subposet p.9/
20 Known asymptotic results V r Thanh(998)/DeBonis, Katona,(006) La(n,V r ) = ( + O( ( ) n n )) n C 4 DeBonis, Katona,Swapepoel,(006) La(n,C 4 ) = ( + O( ( ) n n )) n On families of subsets with a forbidden subposet p.9/
21 Known asymptotic results V r Thanh(998)/DeBonis, Katona,(006) La(n,V r ) = ( + O( ( ) n n )) n C 4 DeBonis, Katona,Swapepoel,(006) La(n,C 4 ) = ( + O( ( ) n n )) n N Griggs, Katona(006) La(n, N ) = ( + O( ( ) n n )) n On families of subsets with a forbidden subposet p.9/
22 Result on P k (s, t) P k (s,t): the blow-up of chain P k by duplicating top element s times and bottom element t times. On families of subsets with a forbidden subposet p.0/
23 Result on P k (s, t) P k (s,t): the blow-up of chain P k by duplicating top element s times and bottom element t times. Theorem(Griggs, Lu007) For s,t and k 3, La(n,P k (s,t)) = (k + O( ( ) n n )) n. On families of subsets with a forbidden subposet p.0/
24 Result on P k (s, t) P k (s,t): the blow-up of chain P k by duplicating top element s times and bottom element t times. Theorem(Griggs, Lu007) For s,t and k 3, La(n,P k (s,t)) = (k + O( ( ) n n )) n. Remark: The hidden constant in O() is a function of s, t, and k. On families of subsets with a forbidden subposet p.0/
25 Result on poset with depth Theorem(Griggs, Lu 007) For any poset H of depth, ( O( ( ) n n )) n La(n,H) ( + O( ( ) n n )) n. On families of subsets with a forbidden subposet p./
26 Result on poset with depth Theorem(Griggs, Lu 007) For any poset H of depth, ( O( ( ) n n )) n La(n,H) ( + O( ( ) n n )) n. Moreover, if H contains the butterfly C 4, then La(n,H) = ( + O( ( ) n n )) n. On families of subsets with a forbidden subposet p./
27 Proof of the upper bound: Since H has depth. Assume H has s elements in upper level and t elements in lower level. H is a subposet of P 3 (s,t). On families of subsets with a forbidden subposet p./
28 Proof of the upper bound: Since H has depth. Assume H has s elements in upper level and t elements in lower level. H is a subposet of P 3 (s,t). We have La(n,H) La(n,P 3 (s,t)) ( + O( ( ) n n )) n. On families of subsets with a forbidden subposet p./
29 Result on up-down tree T is an up-down tree if T is a poset of depth. T is a tree as a graph. On families of subsets with a forbidden subposet p.3/
30 Result on up-down tree T is an up-down tree if T is a poset of depth. T is a tree as a graph. Theorem(Griggs, Lu 007) For any up-down tree T, La(n,T) = ( + O( ( ) n n )) n. On families of subsets with a forbidden subposet p.3/
31 Result on cycles Let C k be the poset of depth on k elements which is also an even cycle as a graph. On families of subsets with a forbidden subposet p.4/
32 Result on cycles Let C k be the poset of depth on k elements which is also an even cycle as a graph. Theorem(Griggs, Lu007) For k 3, ( ) n La(n,C 4k ) = ( + o n ()) n. On families of subsets with a forbidden subposet p.4/
33 Next... We will prove a special case of our theorem here: ( La(n,P 3 (s,t)) + C ) ( ) n n + O(n 3/ ln n) n where C = 6(s + t ). On families of subsets with a forbidden subposet p.5/
34 A probabilistic lemma Lemma: Let X be a random variable taking values of non-negative integers. For any integers k > r, if E(X) > k, then ( ) X E k ( ) X r! E r k! k (E(X) i). i=r On families of subsets with a forbidden subposet p.6/
35 A probabilistic lemma Lemma: Let X be a random variable taking values of non-negative integers. For any integers k > r, if E(X) > k, then ( ) X E k ( ) X r! E r k! k (E(X) i). i=r Special case r = k, ( ) ( ) X X E E (E(X) k + ). k k k On families of subsets with a forbidden subposet p.6/
36 Random variable X σ is a (uniformly) random permutation in S n. On families of subsets with a forbidden subposet p.7/
37 Random variable X σ is a (uniformly) random permutation in S n. C σ is a random chain {σ } {σ,σ } {σ,σ,,σ n }. On families of subsets with a forbidden subposet p.7/
38 Random variable X σ is a (uniformly) random permutation in S n. C σ is a random chain {σ } {σ,σ } {σ,σ,,σ n }. X = F C σ On families of subsets with a forbidden subposet p.7/
39 Random variable X σ is a (uniformly) random permutation in S n. C σ is a random chain X = F C σ {σ } {σ,σ } {σ,σ,,σ n }. E(X) = F F ( n F ). On families of subsets with a forbidden subposet p.7/
40 Random variable X σ is a (uniformly) random permutation in S n. C σ is a random chain {σ } {σ,σ } {σ,σ,,σ n }. X = F C σ E(X) = F F ( n F ). E ( X) 3 = B F ( B ) n A F A B ( B A ) C F B C ( n B n C ). On families of subsets with a forbidden subposet p.7/
41 Random variable X σ is a (uniformly) random permutation in S n. C σ is a random chain {σ } {σ,σ } {σ,σ,,σ n }. X = F C σ E(X) = F F ( n F ). E ( X) 3 = B F ( B ) n E ( X) = B F ( B ) n A F A B A F A B ( B A ) ( B A ). C F B C ( n B n C ). On families of subsets with a forbidden subposet p.7/
42 Random variable X σ is a (uniformly) random permutation in S n. C σ is a random chain {σ } {σ,σ } {σ,σ,,σ n }. X = F C σ E(X) = F F ( n F ). E ( X) 3 = B F ( B ) n E ( X) = B F E ( X ) = ( B ) n ( B ) n B F A F A B A F A B C F B C ( B A ) ( B A ). ( n B n C ). C F B C ( n B n C ). On families of subsets with a forbidden subposet p.7/
43 Reduction A fact on binomial coefficients: ( ) n = O i i n > n ln n ( ( )) n n 3/ n. On families of subsets with a forbidden subposet p.8/
44 Reduction A fact on binomial coefficients: ( ) n = O i i n > n ln n ( ( )) n n 3/ n. F: any P 3 (s,t)-free family of subsets of [n]. On families of subsets with a forbidden subposet p.8/
45 Reduction A fact on binomial coefficients: ( ) n = O i i n > n ln n ( ( )) n n 3/ n. F: any P 3 (s,t)-free family of subsets of [n]. F ( = F \ [n] ) i n > nln n i. On families of subsets with a forbidden subposet p.8/
46 Reduction A fact on binomial coefficients: ( ) n = O i i n > n ln n ( ( )) n n 3/ n. F: any P 3 (s,t)-free family of subsets of [n]. F ( = F \ [n] ) i n > nln n i. ( F F = O n 3/( n ) ) n. On families of subsets with a forbidden subposet p.8/
47 Reduction A fact on binomial coefficients: ( ) n = O i i n > n ln n ( ( )) n n 3/ n. F: any P 3 (s,t)-free family of subsets of [n]. F ( = F \ [n] ) i n > nln n i. ( F F = O n 3/( n ) ) n. Use F instead of F. On families of subsets with a forbidden subposet p.8/
48 Reduction A fact on binomial coefficients: ( ) n = O i i n > n ln n ( ( )) n n 3/ n. F: any P 3 (s,t)-free family of subsets of [n]. F ( = F \ [n] ) i n > nln n i. ( F F = O n 3/( n ) ) n. Use F instead of F. WLOG, we can assume F only contains k-sets with k ( n n ln n, n + n ln n). On families of subsets with a forbidden subposet p.8/
49 ( ) An upper bound of E X 3 For any B F, one of the followings must occur: At most s s A s satisfy A F and A B. A F A B ( B A ) (s ) n n ln n. At most t s C s satisfy C F and B C. C F B C ( n B n C ) (t ) n n ln n. On families of subsets with a forbidden subposet p.9/
50 ( ) An upper bound of E X 3 For any B F, one of the followings must occur: At most s s A s satisfy A F and A B. A F A B ( B A ) (s ) n n ln n. At most t s C s satisfy C F and B C. ( ) X E 3 C F B C ( n B n C ( X E ) (t ) ) (s + t ) n n ln n. n n ln n. On families of subsets with a forbidden subposet p.9/
51 ( ) An upper bound of E X 3 For any B F, one of the followings must occur: At most s s A s satisfy A F and A B. A F A B ( B A ) (s ) n n ln n. At most t s C s satisfy C F and B C. ( ) X E 3 C F B C ( n B n C ( X E ) (t ) ) (s + t ) n n ln n. n n ln n. On families of subsets with a forbidden subposet p.9/
52 Put together ( ) X E 3 ( ) X E (s + t ) n n ln n On families of subsets with a forbidden subposet p.0/
53 Put together ( ) X E 3 ( ) X E (s + t ) ( ) X E 3 3 E ( X n n ln n ) (E(X) ) On families of subsets with a forbidden subposet p.0/
54 Put together ( ) X E 3 ( ) X E (s + t ) ( ) X E 3 3 E ( X (E(X) ) 3(s + t ) n n ln n ) (E(X) ) n n ln n. On families of subsets with a forbidden subposet p.0/
55 Put together ( ) X E 3 ( ) X E (s + t ) ( ) X E 3 3 E ( X (E(X) ) 3(s + t ) n n ln n ) (E(X) ) n n ln n. E(X) + 6 (n n (s + t ) + O 3/ ) ln n. On families of subsets with a forbidden subposet p.0/
56 Put together ( ) X E 3 ( ) X E (s + t ) ( ) X E 3 3 E ( X (E(X) ) 3(s + t ) n n ln n ) (E(X) ) n n ln n. E(X) + 6 (n n (s + t ) + O 3/ ) ln n. The proof is finished since F E(X) ( n n ). On families of subsets with a forbidden subposet p.0/
57 Open problems Let π(h) = lim n La(n,H) ( n ) if it exists. For poset H of depth, we proved π(h) = { π(h). if H is a tree; if H contains C 4. On families of subsets with a forbidden subposet p./
58 Open problems Let π(h) = lim n La(n,H) ( n ) if it exists. For poset H of depth, we proved π(h) = { π(h). if H is a tree; if H contains C 4. Is there a poset H of depth such that < π(h) <? On families of subsets with a forbidden subposet p./
59 Open problems Let π(h) = lim n La(n,H) ( n ) if it exists. For poset H of depth, we proved π(h) = { π(h). if H is a tree; if H contains C 4. Is there a poset H of depth such that < π(h) <? Determine π(c 6 ). On families of subsets with a forbidden subposet p./
60 Open problems Let π(h) = lim n La(n,H) ( n ) if it exists. For poset H of depth, we proved π(h) = { π(h). if H is a tree; if H contains C 4. Is there a poset H of depth such that < π(h) <? Determine π(c 6 ). Determine π(q ). On families of subsets with a forbidden subposet p./
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