### 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 |
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Pages (from-to) | 349-362 |

Number of pages | 14 |

Journal | Order |

Volume | 35 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 Jul 1 |

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### 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