Combinatorics with definable sets: Euler characteristics and Grothendieck rings

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counterexamples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.