Mathematical Sciences: Infinitary Combinatorics and Social Choice

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Taylor's investigations will all be in the area of combinatorial set theory, but the questions to be considered have rather different inspirations. That is, those of the first group to be discussed arise from what one thinks of as traditional infinitary combinatorics, while those of the second group arise from the subarea of mathematical political science launched into prominence by the Nobel Prize winning work of Kenneth Arrow. The considerations within traditional infinitary combinatorics will begin with a specific question about definable graphs on the real line. The result it is hoped to establish for all Borel sets has been shown to hold for closed sets by M. Foreman and the investigator; the desired extension would provide a simultaneous generalization of two elegant known results. The more general investigations will be in the area of partition theory and primarily involve some hoped for extensions of recent joint work with A. Blass and P. Erdos. Arrow's Impossibility Theorem can be regarded as an elegant characterization of the ultraproduct construction applied to linear (or pre-linear) orderings. (Since ultrafilters on a finite set are principal, this yields the "dictator" one associates with Arrow's theorem). The instant investigations in social choice theory will focus on the investigator's recent work generalizing Arrow's theorem in both the finite context and that of ultra-products.