## Abstract

Given a finite poset *P*, the intensively studied quantity L*a*(*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 \({\mathrm La}(n,P)\le \frac {1}{2}(|P|+h-2)\left (\begin {array}{c}n \\ \lfloor {n/2}\rfloor \end {array}\right )\) for any finite poset *P* of height *h*. This paper elaborates their double-chain method to obtain a new upper bound

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).

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The authors would like to thank the anonymous reviewers for their valuable comments to improve the organization and presentation of this paper.

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## Additional information

Jun-Yi Guo research was supported by MOST-104-2115-M-003-010.

Fei-Huang Chang research was supported by MOST-104-2115-M-003-008-MY2.

Hong-Bin Chen research was supported by MOST-105-2115-M-035-006-MY2.

Wei-Tian Li research was supported by MOST-103-2115-M-005-003-MY2.

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Guo, JY., Chang, FH., Chen, HB. *et al.* Families of Subsets Without a Given Poset in Double Chains and Boolean Lattices.
*Order* **35**, 349–362 (2018). https://doi.org/10.1007/s11083-017-9436-1

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DOI: https://doi.org/10.1007/s11083-017-9436-1