Coast to Coast Seminar Series: Live from Winnipeg, Manitoba "Ramsey Theory and the Infinite"

Abstract

This talk is an invitation to infinite Ramsey theory, accessible to most mathematicians. Most combinatorists are familiar with Ramsey theory regarding finite structures, and many are aware of some infinitary techniques often used to solve Ramsey questions in the finite, for example, ultrafilters, harmonic analysis, and ergodic techniques. However, it seems that few combinatorists are familiar with much infinite Ramsey theory. On the other hand, topologists, analysts, and set theorists seem to regularly use Ramsey theory in the infinite, but it seems that only a few basic theorems find application. I survey some infinite Ramsey-type theorems (with few or no proofs) and hope to reveal some surprises. One surprise to me is that, often, topology is required to refine an infinite Ramsey-type statement before finding a proof for that statement. My expertise is not infinite Ramsey theory, and I claim no expertise in topology, but I hope to bring the audience to the point where it is clear that topology might help, or even be required, to further advance the field of Ramsey theory. For those not familiar with Ramsey theory, a typical theorem has the form: for any r (number of colours), H (small structure or set) and G (medium), there exists a (large) F so that for any r-colouring of the H-substructures of F, there exists a G-substructure in F all of whose H-substructures are monochromatic. For example, the pigeonhole principle is such a theorem (where H is a single vertex). In Ramsey's original theorem, r is finite, H, G, and F are simply sets, where H is finite, and G is finite or countably infinite.