### Abstract

Given a finite poset P, the intensively studied quantity La(n, P) denotes the largest size of a family of subsets of [n] not containing P as a weak subposet. Burcsi and Nagy (J. Graph Theory Appl. 1, 40–49 2013) proposed a double-chain method to get an upper bound La(n,P)≤12(|P|+h−2)(n⌊n/2⌋) for any finite poset P of height h. This paper elaborates their double-chain method to obtain a new upper boundLa(n,P)≤(|P|+h−α(GP)−22)(n⌊n2⌋)for any graded poset P, where α(G_{P}) denotes the independence number of an auxiliary graph defined in terms of P. This result enables us to find more posets which verify an important conjecture by Griggs and Lu (J. Comb. Theory (Ser. A) 119, 310–322, 2012).

Original language | English |
---|---|

Pages (from-to) | 349-362 |

Number of pages | 14 |

Journal | Order |

Volume | 35 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 Jul 1 |

### Fingerprint

### Keywords

- Double counting
- Extremal family
- Graded poset
- Interval chains
- Poset-free families

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics

### Cite this

*Order*,

*35*(2), 349-362. https://doi.org/10.1007/s11083-017-9436-1

**Families of Subsets Without a Given Poset in Double Chains and Boolean Lattices.** / Guo, Jun-Yi; Chang, Fei Huang; Chen, Hong Bin; Li, Wei Tian.

Research output: Contribution to journal › Article

*Order*, vol. 35, no. 2, pp. 349-362. https://doi.org/10.1007/s11083-017-9436-1

}

TY - JOUR

T1 - Families of Subsets Without a Given Poset in Double Chains and Boolean Lattices

AU - Guo, Jun-Yi

AU - Chang, Fei Huang

AU - Chen, Hong Bin

AU - Li, Wei Tian

PY - 2018/7/1

Y1 - 2018/7/1

N2 - Given a finite poset P, the intensively studied quantity La(n, P) denotes the largest size of a family of subsets of [n] not containing P as a weak subposet. Burcsi and Nagy (J. Graph Theory Appl. 1, 40–49 2013) proposed a double-chain method to get an upper bound La(n,P)≤12(|P|+h−2)(n⌊n/2⌋) for any finite poset P of height h. This paper elaborates their double-chain method to obtain a new upper boundLa(n,P)≤(|P|+h−α(GP)−22)(n⌊n2⌋)for any graded poset P, where α(GP) denotes the independence number of an auxiliary graph defined in terms of P. This result enables us to find more posets which verify an important conjecture by Griggs and Lu (J. Comb. Theory (Ser. A) 119, 310–322, 2012).

AB - Given a finite poset P, the intensively studied quantity La(n, P) denotes the largest size of a family of subsets of [n] not containing P as a weak subposet. Burcsi and Nagy (J. Graph Theory Appl. 1, 40–49 2013) proposed a double-chain method to get an upper bound La(n,P)≤12(|P|+h−2)(n⌊n/2⌋) for any finite poset P of height h. This paper elaborates their double-chain method to obtain a new upper boundLa(n,P)≤(|P|+h−α(GP)−22)(n⌊n2⌋)for any graded poset P, where α(GP) denotes the independence number of an auxiliary graph defined in terms of P. This result enables us to find more posets which verify an important conjecture by Griggs and Lu (J. Comb. Theory (Ser. A) 119, 310–322, 2012).

KW - Double counting

KW - Extremal family

KW - Graded poset

KW - Interval chains

KW - Poset-free families

UR - http://www.scopus.com/inward/record.url?scp=85028777159&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028777159&partnerID=8YFLogxK

U2 - 10.1007/s11083-017-9436-1

DO - 10.1007/s11083-017-9436-1

M3 - Article

AN - SCOPUS:85028777159

VL - 35

SP - 349

EP - 362

JO - Order

JF - Order

SN - 0167-8094

IS - 2

ER -