By Daniel J. Velleman

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**Extra resources for American Mathematical Monthly, volume 117, number 1, january 2010**

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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}.

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