Model Theory Seminar

October 4
Alexander Van Abel, CUNY
On Pseudofinite Dimension and Measure
A pseudofinite structure is (among many equivalent definitions) a structure which is elementarily equivalent to an infinite ultraproduct of finite structures. In this talk, we discuss how the natural counting measure on finite structures lifts to useful notions of dimension and measure on pseudofinite structures. We give a proof of Furstenburg's Correspondence Principle in combinatorial number theory, using pseudofinite measure. We sketch a proof, by Chernikov and Starchenko, of a special case ('stable' graphs) of the Erdös-Hajnal conjecture, using a particular notion of pseudofinite dimension. Finally, we discuss how a different notion of dimension leads to simplicity results. This talk is largely based on Darío García's lecture notes on 'Model theory of pseudofinite structures'.