Set Families with a Forbidden Subposet

Bukh, Boris

doi:10.37236/231

Cited by 63 publications

(108 citation statements)

References 7 publications

Supporting

Mentioning

106

Contrasting

Order By: Relevance

“…A remarkable result concerning Conjecture 1.1 is that of Bukh [2], who verified Conjecture 1.1 (i) for tree posets. In the following results we strengthen his result in two cases.…”

Section: Asymptotic Resultsmentioning

confidence: 92%

See 1 more Smart Citation

Forbidding Rank-Preserving Copies of a Poset

Gerbner

Methuku²,

Nagy

et al. 2019

Order

View full text Add to dashboard Cite

The maximum size, La(n, P ), of a family of subsets of [n] = {1, 2, ..., n} without containing a copy of P as a subposet, has been intensively studied.Let P be a graded poset. We say that a family F of subsets of [n] = {1, 2, ..., n} contains a rank-preserving copy of P if it contains a copy of P such that elements of P having the same rank are mapped to sets of same size in F. The largest size of a family of subsets of [n] = {1, 2, ..., n} without containing a rank-preserving copy of P as a subposet is denoted by La rp (n, P ). Clearly, La(n, P ) ≤ La rp (n, P ) holds.In this paper we prove asymptotically optimal upper bounds on La rp (n, P ) for tree posets of height 2 and monotone tree posets of height 3, strengthening a result of Bukh in these cases. We also obtain the exact value of La rp (

show abstract

“…A remarkable result concerning Conjecture 1.1 is that of Bukh [2], who verified Conjecture 1.1 (i) for tree posets. In the following results we strengthen his result in two cases.…”

Section: Asymptotic Resultsmentioning

confidence: 92%

“…The proof of Theorem 1.2 follows the lines of a reasoning of Bukh's [2]. The new idea is that we count the number of related pairs between two fixed levels as detailed in the proof below.…”

Section: Proof Of Theorem 12: Trees Of Height Twomentioning

confidence: 99%

Forbidding Rank-Preserving Copies of a Poset

Gerbner

Methuku²,

Nagy

et al. 2019

Order

View full text Add to dashboard Cite

show abstract

“…Let I i 1 be the family of light elements of [k/s i ] d , I i 2 be the family of fat elements, and I i 3 be the family of medium elements. Also, for j ∈ [3],…”

Section: Finding P In Sparse Familiesmentioning

confidence: 99%

“…The value of La(n, P ) was also studied for a number of fixed posets such as forks and brooms [7,19,28], diamond [14,22], butterfly [8], cycles C 4k on two levels [15]. In case the Hasse diagram of P is a tree, it was proved by Bukh [3] that La(n, P ) < (h − 1 + o(1)) n ⌊n/2⌋ , where h is the height of P , and Boehnlein and Jiang [2] improved this result by showing that La # (n, P ) < (h − 1 + o(1)) n ⌊n/2⌋ also holds.…”

Section: Introductionmentioning

confidence: 99%

Forbidden induced subposets of given height

Tomon

2019

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

Let P be a partially ordered set. The function La # (n, P ) denotes the size of the largest family F ⊂ 2 [n] that does not contain an induced copy of P . It was proved by Methuku and Pálvölgyi that there exists a constant CP (depending only on P ) such that La # (n, P ) < CP n ⌊n/2⌋ . However, the order of the constant CP following from their proof is typically exponential in |P |. Here, we show that if the height of the poset is constant, this can be improved. We show that for every positive integer h there exists a constant c h such that if P has height at most h, thenOur methods also immediately imply that similar bounds hold in grids as well. That is, we show that if F ⊂ [k] n such that F does not contain an induced copy of P and n ≥ 2|P |, thenA small part of our proof is to partition 2 [n] (or [k] n ) into certain fixed dimensional grids of large sides. We show that this special partition can be used to derive bounds in a number of other extremal set theoretical problems and their generalizations in grids, such as the size of families avoiding weak posets, Boolean algebras, or two distinct sets and their union. This might be of independent interest.

show abstract

“…A poset is called a tree poset if its Hasse diagram is a tree. Bukh [1] generalized Theorem 1.2 to all tree posets by showing that for any tree poset T of height h(T ), we have La(n, T ) = (h(T ) − 1 + o(1)) n ⌊ n 2 ⌋ . Recently Gerbner, Keszegh and Patkós [3] initiated the investigation of counting the maximum number of copies of a poset in a family F ⊆ 2 [n] that is P-free.…”

Section: Introductionmentioning

confidence: 99%

On the Number of Containments in P-free Families

Gerbner

Methuku

Nagy

et al. 2019

Graphs and Combinatorics

View full text Add to dashboard Cite

A subfamily {F 1 , F 2 , . . . , F |P | } ⊆ F is a copy of the poset P if there exists a bijection i : P → {F 1 , F 2 , . . . , F |P | } such that p ≤ P q implies i(p) ⊆ i(q). A family F is P -free, if it does not contain a copy of P . In this paper we establish basic results on the maximum number of k-chains in a P -free family F ⊆ 2 [n] . We prove that if the height of P , h(P ) > k, then this number is of the order Θ( k+1 i=1 l i−1 l i ), where l 0 = n and l 1 ≥ l 2 ≥ · · · ≥ l k+1 are such that n − l 1 , l 1 − l 2 , . . . , l k − l k+1 , l k+1 differ by at most one. On the other hand if h(P ) ≤ k, then we show that this number is of smaller order of magnitude.Let ∨ r denote the poset on r + 1 elements a, b 1 , b 2 , . . . , b r , where a < b i for all 1 ≤ i ≤ r and let ∧ r denote its dual. For any values of k and l, we construct a {∧ k , ∨ l }-free family and we conjecture that it contains asymptotically the maximum number of pairs in containment.We prove that this conjecture holds under the additional assumption that a chain of length 4 is forbidden. Moreover, we prove the conjecture for some small values of k and l. We also derive the asymptotics of the maximum number of copies of certain tree posets T of height 2 in {∧ k , ∨ l }-free families F ⊆ 2 [n] .

show abstract

Set Families with a Forbidden Subposet

Abstract: We asymptotically determine the size of the largest family F of subsets of {1, . . . , n} not containing a given poset P if the Hasse diagram of P is a tree. This is a qualitative generalization of several known results including Sperner's theorem.

Cited by 63 publications

References 7 publications

Forbidding Rank-Preserving Copies of a Poset

Forbidding Rank-Preserving Copies of a Poset

Forbidden induced subposets of given height

On the Number of Containments in P-free Families

Contact Info

Product

Resources

About