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We examine the topological dynamics of the automorphism groups ofomega-categorical sparse graphs resulting from Hrushovski constructions.Specifically, we consider the fixed points on type spaces property, which astructure M has if, for each positive integer n, every Aut(M)-subflow of thespace of n-types has a fixed point. Extending a result of Evans, Hubicka andNesetril, we show that there exists an omega-categorical structure M, resultingfrom a Hrushovski construction, such that no omega-categorical expansion of Mhas the fixed points on type spaces property.

Sparse graphs and the fixed points on type spaces property