Geometry & Topology Seminar: "Szemeredi's theorem on arithmetic progressions I: ergodic theory approach"

Abstract: Given natural numbers c and k there exists a natural number W such that any coloring of the first W natural numbers with c colors contains a monochromatic arithmetic progression with k terms (van der Waerden, 1927). More strongly, given natural number k and positive real number a there exists a natural number N such that any subset of the first N natural numbers with cardinality no less than aN contains an arithmetic progression with k terms (Szemeredi, 1975). We explain Furstenberg's ergodic theory approach to deduce these theorems. In an Analysis Seminar (October 30) we present Roth's Fourier analysis approach to prove Szemeredi's theorem when k=3.

Host: Xiang Tang