Sparse graphs and the fixed points on type spaces property

Sparse graphs and the fixed points on type spaces property Rob SullivanWe examine the topological dynamics of the automorphism groups of ωcategorical sparse graphs resulting from Hrushovski constructions.Specifically, we consider the fixed points on type spaces property, which a structure M has if, for all n ∈ ‫,ގ‬ every Aut(M)-subflow of the space S n (M) of n-types has a fixed point.Extending a result of Evans, Hubička and Nešetřil, we show that there exists an ω-categorical structure M, resulting from a Hrushovski construction, such that no ω-categorical expansion of M has the fixed points on type spaces property.We recall that, for a Hausdorff topological group G, a G-flow is a continuous action of G on a nonempty compact Hausdorff space X , and we say that G is extremely amenable if every G-flow has a G-fixed point.In this paper, we show that the above result holds even in the context of a more restricted class of flows: subflows of type spaces.Let M be a relational structure.Following Meir and Sullivan 2023, we say that M has the fixed points on type spaces property (FPT), if, for each n ∈ ‫ގ‬ + , every subflow of S n (M) has an Aut(M)fixed point, where S n (M) denotes the Stone space of n-types with parameters from M and the action is given by translation of parameters in formulae.This property is studied in depth in loc.cit., and may be thought of as a restriction of extreme amenability to a subclass of flows which occur naturally in a model-theoretic context.