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Daniel J. Velleman's American Mathematical Monthly, volume 117, number 1, january PDF

By Daniel J. Velleman

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

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In approval voting systems, each voter indicates approval or disapproval for each candidate or option (we use the term platform), and the platform with the most approval votes wins. What conditions guarantee that there will be a platform with the approval of at least half (or all) the voters? Without introducing any further assumptions, one can only reach trivial conclusions, but in many situations, there is some underlying property of the set of platforms and votes that allows for more interesting results.

N}, I is good, and |J | = m. Let g be the number of good sets. Since every good set is of size d + 1 and extends to an m-element n−d−1 n−d−1 subset of {1, 2, . . , n} in m−d−1 ways, we have N = g m−d−1 . On the other hand, every m-element set J ⊆ {1, 2, . . , n} includes at least one k-element set K with k i∈K Ai = ∅ (because S is (k, m)-agreeable), and K in turn includes d+1 good sets. k n k n m , and hence g ≥ d+1 , as desired. Thus N ≥ d+1 m d+1 d+1 For d = 1, Theorem 10 gives a worse bound than Theorem 8, and hence for d ≥ 2, the bound is most likely not best possible.

Since C1 ∪ C2 ∪ · · · ∪ Ck−1 is colored using k − 1 colors, it includes no clique of size k, and hence, |C1 ∪ C2 ∪ · · · ∪ Ck−1 | ≤ m − 1. January 2010] VOTING IN AGREEABLE SOCIETIES 33 It follows that |Ck−1 | ≤ q, for otherwise |C1 ∪ C2 ∪ · · · ∪ Ck−1 | ≥ (k − 1)(q + 1) ≥ (k − 1)q + ρ + 1 = m, a contradiction. Thus |Ci | ≤ q for each i ≥ k and χ k−1 |Ci | + n= i=1 |Ci | ≤ m − 1 + (χ − k + 1)q i=k = (k − 1)q + ρ + (χ − k + 1)q = χq + ρ, as desired. Theorem 8. Let 2 ≤ k ≤ m. If G is the agreement graph of a (k, m)-agreeable linear society and q, ρ are defined by the division with remainder: m − 1 = (k − 1)q + ρ, ρ ≤ k − 2, then the clique number satisfies: ω(G) ≥ n−ρ , q and this bound is best possible.

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American Mathematical Monthly, volume 117, number 1, january 2010 by Daniel J. Velleman


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