Abstract
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.
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Brandt, F., Chudnovsky, M., Kim, I. et al. A counterexample to a conjecture of Schwartz. Soc Choice Welf 40, 739–743 (2013). https://doi.org/10.1007/s00355-011-0638-y
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DOI: https://doi.org/10.1007/s00355-011-0638-y