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

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

$${\mathrm La}(n,P)\le \left( \frac{|P|+h-\alpha(G_{P})-2}{2}\right)\left( \begin{array}{c}n \\ \lfloor{\frac{n}{2}}\rfloor\end{array}\right) $$

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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Acknowledgments

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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Authors and Affiliations

  1. Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan

    Jun-Yi Guo

  2. Division of Preparatory Programs for Overseas Chinese Students, National Taiwan Normal University, New Taipei, 24449, Taiwan

    Fei-Huang Chang

  3. Department of Applied Mathematics, Feng Chia University, Taichung, 40724, Taiwan

    Hong-Bin Chen

  4. Department of Applied Mathematics, National Chung Hsing University, Taichung, 40227, Taiwan

    Wei-Tian Li

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  1. Jun-Yi Guo
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  2. Fei-Huang Chang
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  4. Wei-Tian Li
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Corresponding author

Correspondence to Wei-Tian Li.

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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