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On the Union of Arithmetic Progressions
We show that for every there is an absolute constant
such that the following is true. The union of any
arithmetic progressions, each of length , with pairwise distinct differences
must consist of at least elements. We
observe, by construction, that one can find arithmetic progressions, each
of length , with pairwise distinct differences such that the cardinality of
their union is . We refer also to the non-symmetric case of
arithmetic progressions, each of length , for various regimes of and
Majority vote following a debate
Voters determine their preferences over alternatives based on cases (or arguments) that are raised in the public debate. Each voter is characterized by a matrix, measuring how much support each case lends to each alternative, and her ranking is additive in cases. We show that the majority vote in such a society can be any function from sets of cases to binary relations over alternatives. A similar result holds for voting with quota in the case of two alternatives.Case-based decision theory; voting theory; debates
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