Saturating Sperner Families

Gerbner, Dániel; Keszegh, Balázs; Lemons, Nathan; Palmer, Cory; Pálvölgyi, Dömötör; Patkós, Balázs

doi:10.1007/s00373-012-1195-6

Cited by 35 publications

(51 citation statements)

References 23 publications

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“…Let P k denote the k-element chain, which is the poset with k elements in which each pair of elements is comparable. A main result of [6] is the following.…”

Section: Introductionmentioning

confidence: 98%

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The saturation number of induced subposets of the Boolean lattice

Ferrara

Kay

Kramer

et al. 2017

Discrete Mathematics

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Given a poset P, a family F of elements in the Boolean lattice is said to be P-saturated if (1) F contains no copy of P as a subposet and (2) every proper superset of F contains a copy of P as a subposet. The maximum size of a P-saturated family is denoted by La(n, P), which has been studied for a number of choices of P. The minimum size of a P-saturated family, sat(n, P), was introduced by Gerbner et al. (2013), and parallels the deep literature on the saturation function for graphs.We introduce and study the concept of saturation for induced subposets. As opposed to induced saturation in graphs, the above definition of saturation for posets extends naturally to the induced setting. We give several exact results and a number of bounds on the induced saturation number for several small posets. We also use a transformation to the biclique cover problem to prove a logarithmic lower bound for a rich infinite family of target posets. †

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“…Let P k denote the k-element chain, which is the poset with k elements in which each pair of elements is comparable. A main result of [6] is the following.…”

Section: Introductionmentioning

confidence: 98%

“…Gerbner et al [6] introduced the saturation function in the setting of posets. Let P k denote the k-element chain, which is the poset with k elements in which each pair of elements is comparable.…”

Section: Introductionmentioning

confidence: 99%

The saturation number of induced subposets of the Boolean lattice

Ferrara

Kay

Kramer

et al. 2017

Discrete Mathematics

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“…Construction 2 (Gerbner et al [11]). Let Y be a set such that |Y | = k − 2 and let H be a non-empty set disjoint from Y .…”

Section: Introductionmentioning

confidence: 99%

“…Given a set X, a collection F ⊆ P(X) is said to be k-Sperner if it does not contain a chain of length k + 1 under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al [11] conjectured that, if |X| is sufficiently large with respect to k, then the minimum size of a saturated k-Sperner system F ⊆ P(X) is 2 k−1 . We disprove this conjecture by showing that there exists ε > 0 such that for every k and |X| ≥ n 0 (k) there exists a saturated k-Sperner system F ⊆ P(X) with cardinality at most 2 (1−ε)k .…”

mentioning

confidence: 99%

“…A collection F ⊆ P(X) is said to be an oversaturated k-Sperner system if, for every S ∈ P(X) \ F, F ∪ {S} contains more chains of length k + 1 than F. Gerbner et al [11] proved that, if |X| ≥ k, then the smallest such collection contains between 2 k/2−1 and O log k k 2 k elements. We show that if |X| ≥ k 2 + k, then the lower bound is best possible, up to a polynomial factor.…”

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confidence: 99%

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On Saturated $k$-Sperner Systems

Morrison

Noel

Scott

2014

Electron. J. Combin.

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Given a set X, a collection F ⊆ P(X) is said to be k-Sperner if it does not contain a chain of length k + 1 under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. [11] conjectured that, if |X| is sufficiently large with respect to k, then the minimum size of a saturated k-Sperner system F ⊆ P(X) is 2 k−1 . We disprove this conjecture by showing that there exists ε > 0 such that for every k and |X| ≥ n 0 (k) there exists a saturated k-Sperner system F ⊆ P(X) with cardinality at most 2 (1−ε)k .A collection F ⊆ P(X) is said to be an oversaturated k-Sperner system if, for every S ∈ P(X) \ F, F ∪ {S} contains more chains of length k + 1 than F. Gerbner et al. [11] proved that, if |X| ≥ k, then the smallest such collection contains between 2 k/2−1 and O log k k 2 k elements. We show that if |X| ≥ k 2 + k, then the lower bound is best possible, up to a polynomial factor.

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