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It is not (3, 4)agreeable, however, because among voters 1, 2, 4, 7 no three of them share a common platform. The same society, after restricting the spectrum to a set of candidates, is the linear society shown in Figure 3. It is not (2, 3)-agreeable, because among voters 2, 4, 7 there is no pair that can agree on a candidate (in fact, voter 7 does not approve any candidate). However, one may verify that this linear society is (2, 4)-agreeable. For a society S, the agreement number of a platform, a( p), is the number of voters in S who approve of platform p.

A. , Perfect Graphs, Wiley, Chichester, UK, 2001. 19. C. E. E. Trans. Inform. Theory IT-2 (1956) 8–19. 1056798 20. D. B. West, Introduction to Graph Theory, Prentice-Hall, Upper Saddle River, NJ, 2001. 21. M. Yannakakis, The complexity of the partial order dimension problem, SIAM J. Algebraic Discrete Methods 3 (1982) 351–358. 1137/0603036 DEBORAH E. S. S. in Mathematics from the University of Nebraska-Lincoln in 2008. D. in Mathematics Education. When not busy studying, doing graph theory, or teaching, she spends her time practicing Shotokan karate.

10. C. S. Hardin, Agreement in circular societies, Amer. Math. Monthly 117 (2010) 40–49. 4169/ 000298910X474970 38 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 11. G. Kalai, Intersection patterns of convex sets, Israel J. Math. 48 (1984) 161–174. 1007/ BF02761162 12. M. Katchalski and D. Nashtir, A Helly type conjecture, Discrete Comput. Geom. 21 (1999) 37–43. 1007/PL00009408 13. B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes f¨ur n-dimensionale Simplexe, Fund.

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