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Acyclic sets of linear orders: A progress report

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  • Peter C. Fishburn

    (AT&T Shannon Laboratory, Florham Park, NJ 07932, USA)

Abstract

Let f(n) be the maximum cardinality of an acyclic set of linear orders on {1, 2, \dots , n}. It is known that f(3)=4, f(4)=9, f(5)=20, and that all maximum-cardinality acyclic sets for n\leq 5 are constructed by an "alternating scheme". We outline a proof that this scheme is optimal for n=6, where f (6)=45. It is known for large n that f (n) >(2.17)n and that no maximum-cardinality acyclic set conforms to the alternating scheme. Ran Raz recently proved that f (n) 0 and all n. We conjecture that f (n + m)\leqf (n + 1) f (m + 1) for n , m\geq 1, which would imply f (n)

Suggested Citation

  • Peter C. Fishburn, 2002. "Acyclic sets of linear orders: A progress report," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(2), pages 431-447.
    • Handle: RePEc:spr:sochwe:v:19:y:2002:i:2:p:431-447
    Note: Received: 12 April 2000/Accepted: 4 December 2000
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    Cited by:

    1. Gilbert Laffond & Jean Lainé, 2014. "Triple-consistent social choice and the majority rule," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 784-799, July.
    2. Puppe, Clemens, 2018. "The single-peaked domain revisited: A simple global characterization," Journal of Economic Theory, Elsevier, vol. 176(C), pages 55-80.
    3. Liu, Peng, 2020. "Random assignments on sequentially dichotomous domains," Games and Economic Behavior, Elsevier, vol. 121(C), pages 565-584.
    4. Slinko, Arkadii, 2019. "Condorcet domains satisfying Arrow’s single-peakedness," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 166-175.
    5. Salvador Barberà & Dolors Berga & Bernardo Moreno, 2020. "Arrow on domain conditions: a fruitful road to travel," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 54(2), pages 237-258, March.
    6. Li, Guanhao, 2023. "A classification of peak-pit maximal Condorcet domains," Mathematical Social Sciences, Elsevier, vol. 125(C), pages 42-57.
    7. Clemens Puppe & Arkadii Slinko, 2019. "Condorcet domains, median graphs and the single-crossing property," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(1), pages 285-318, February.
    8. Alexander Karpov & Arkadii Slinko, 2023. "Constructing large peak-pit Condorcet domains," Theory and Decision, Springer, vol. 94(1), pages 97-120, January.
    9. Alexander Karpov & Klas Markstrom & S{o}ren Riis & Bei Zhou, 2023. "Bipartite peak-pit domains," Papers 2308.02817, arXiv.org, revised Jan 2024.

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