MS-measurability and omega-categorical Hrushovski constructions – Paolo Marimon

 

A structure is MS-measurable if it admits a dimension-measure function on their definable sets satisfying certain definability, additivity and Fubini conditions. MS-measurable structures are necessarily supersimple of finite rank and (functionally) unimodular. Elwes and Macpherson (2006) asked whether the converse is true. In that paper the authors suggested omega-categorical Hrushovski constructions as a possible source of counterexamples. We are also interested in the question of whether any omega-categorical Hrushovski construction is MS-measurable since they may provide a counterexample to the conjecture in the same paper that any omega-categorical MS-measurable structure is one-based.  

Results from Evans and the first year of my PhD show how in various omega-categorical Hrushovski constructions any dimension satisfying certain weak conditions must be a scalar multiple of the natural notion of dimension in a Hrushovski construction. Evans’ results already gave some counterexamples to Elwes’ and Macpherson’s first question. But those methods cannot be generalised for showing in general that Hrushovski constructions are not MS-measurable. Hence, I’ve been looking at different methods for showing there is no dimension-measure function on a Hrushovski construction with the natural Hrushovski dimension. I recently proved that there is no such dimension-measure function for a class of omega-categorical Hrushovski constructions which are supersimple, finite rank and unimodular graphs. This should yield a more general strategy for showing non-MS-measurability of omega-categorical Hrushovski constructions.