A counterexample to a conjecture of Schwartz

Felix Brandt, Maria Chudnovsky, Ilhee Kim, Gaku Liu, Sergey Norin, Alex Scott, Paul Seymour, Stephan Thomassé

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14 Scopus citations


In 1990, motivated by applications in the social sciences, Thomas Schwartz made a conjecture about tournaments which would have had numerous attractive consequences. In particular, it implied that there is no tournament with a partition A, B of its vertex set, such that every transitive subset of A is in the out-neighbour set of some vertex in B, and vice versa. But in fact there is such a tournament, as we show in this article, and so Schwartz' conjecture is false. Our proof is non-constructive and uses the probabilistic method.

Original languageEnglish (US)
Pages (from-to)739-743
Number of pages5
JournalSocial Choice and Welfare
Issue number3
StatePublished - 2013

All Science Journal Classification (ASJC) codes

  • Social Sciences (miscellaneous)
  • Economics and Econometrics

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    Brandt, F., Chudnovsky, M., Kim, I., Liu, G., Norin, S., Scott, A., Seymour, P., & Thomassé, S. (2013). A counterexample to a conjecture of Schwartz. Social Choice and Welfare, 40(3), 739-743. https://doi.org/10.1007/s00355-011-0638-y