By Daniel J. Velleman
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Additional resources for American Mathematical Monthly, volume 117, number 1, january 2010
Assume that there is a finite set V of voters, and each voter v has an approval set Av of platforms. 28 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 We define a society S to be a triple (X, V, A) consisting of a spectrum X , a set of voters V , and a collection A of approval sets for all the voters. Of particular interest to us will be the case of a linear society, in which X is a closed subset of R and approval sets in A are of the form X ∩ I where I is either empty or a closed bounded interval in R.
2 applies to such societies, and consider a (k, m)agreeable circular society S in which i voters have empty approval sets (call them implacable). Form the society S by throwing away these voters; S must be (k, m − i)agreeable: given m − i voters in S , combine them with the i implacable voters in S; by the (k, m)-agreeability of S, there must be a platform that k of these voters approve of, and each of these voters is in S . ) Since S is (k, m − i)-agreeable, and noting that a(S) = a(S ), we have a(S ) |S | a(S) = · |S| |S | |S| > k − 1 |S | · m − i |S| = m |S| − i k−1 · · m m−i |S| ≥ k−1 m (since |S| ≥ m).
A linear (4, 15)-society with n = 21 voters. Here q = 4 and ρ = 2, so the clique number is at least (n − ρ)/q = 5. The Agreeable Linear Society Theorem (Theorem 2) now follows as a corollary of Theorem 8. 6. Rd -CONVEX AND d-BOX SOCIETIES. In this section we prove a higherdimensional analogue of Theorem 8 by giving a lower bound on the agreement proportion of a (k, m)-agreeable Rd -convex society. We need a different method than our method for d = 1, because for d ≥ 2, neither Fact 1 nor Fact 2 holds.
American Mathematical Monthly, volume 117, number 1, january 2010 by Daniel J. Velleman