Anton Bernshteyn: From finite combinatorics to descriptive set theory and back

Mathematical logic seminar – Sep 4 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Anton Bernshteyn
Department of Mathematical Sciences
CMU

Title:     From finite combinatorics to descriptive set theory and back

Abstract:

Many results in finite combinatorics can be extended to infinite structures via compactness – but this transfer is powered by the Axiom of Choice and leads, in general, to highly “pathological” objects. It is natural to ask, which combinatorial constructions can be performed in a “well-behaved” fashion, say, in a Borel or measurable way? This question is addressed in a young branch of descriptive set theory called descriptive combinatorics. We will discuss a class of coloring problems with the requirement that the desired coloring be Baire measurable (i.e., “topologically well-behaved”). The central result of this talk is that the existence of a Baire measurable coloring is equivalent to a purely combinatorial statement, analogs of which have for a long time been studied in finite graph theory with no relation to descriptive set theory.

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