American Mathematical Monthly, volume 117, number 1, january by Daniel J. Velleman

By Daniel J. Velleman

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Then A y ⊆ A x for some x = y, and (after rotating if necessary) the string rep46 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 resentation of S contains x ρ y σ y τ x , where ρ, σ , and τ are (possibly empty) strings of intervening endpoints. Let R be the society that results from swapping x and y ; that is, we replace x ρ y σ y τ x with x ρ y σ x τ y . ) Note that this only affects the approval sets of x and y. We wish to show S R (although R will not necessarily be the promised society T ).

Now suppose k ≤ mh/N . Choose any set W of m voters in U (N , h). The average value of aW ( p) is 1 N N aW ( p) dp = 0 = N 1 N x∈W 1 N x∈W a{x} ( p) dp 0 h = mh/N . It follows that aW ( p) ≥ mh/N for some platform p; since aW ( p) is an integer, aW ( p) ≥ mh/N ≥ k. So U (N , h) is (k, m)-agreeable. This establishes (iii) ⇒ (i). Finally, to establish (i) ⇒ (iii), it suffices to exhibit a set W of m voters from U (N , h) such that at most mh/N of them can agree on a single platform. Define v : Z → Z by v(i) = i N /m , and let W = {v(i) | 0 ≤ i < m}.

14. J. Kratochvil, A special planar satisfiability problem and some consequences of its NP-completeness, Discrete Appl. Math. 52 (1994) 233–252. 1016/0166-218X(94)90143-0 15. J. Matousek, Lectures on Discrete Geometry, Springer-Verlag, New York, 2002. 16. A. Niedermaier, D. Rizzolo, and F. E. 0339. 17. J. Radon, Mengen konvexer K¨orper, die einen gemeinsamen Punkt enthalten, Math. Ann. 83 (1921) 113– 115. 1007/BF01464231 18. J. L. Ram´ırez-Alfons´ın and B. A. , Perfect Graphs, Wiley, Chichester, UK, 2001.

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