Abstract: 

Joint work with Samuel Braunfeld and Colin Jahel. We study invariant random expansions of a structure $M$ by some hereditary class of structures. For example, we study $\mathrm{Aut}(M)$-invariant (regular) Borel probability measures on the space of graphs on the domain of $M$. We prove that for many homogeneous structures of arity $>2$, all invariant random expansions of $M$ by graphs are exchangeable, i. e. invariant under all permutations. The latter are well-understood due to the Aldous-Hoover theorem in probability. More generally, we define a measure of randomness of $M$, $k$-overlap closedness, which implies that invariant random expansions by hereditary classes of sufficiently slow growth rate, i.e. $O(e^{n^{k+\delta}})$ for all $\delta>0$, are exchangeable. Since invariant Keisler measures are a special case of invariant random expansions, our work allows us to describe the spaces of invariant Keisler measures of various homogeneous structures of arity $>2$. In particular, we show there are $2^{\aleph_0}$ supersimple homogeneous ternary structures for which there are non-forking formulas which are universally measure zero.

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