Algebra Seminar talk

Paolo Marimon
Measures in higher arity homogeneous structures

This is joint work with Samuel Braunfeld and Colin Jahel. We study invariant Keisler measures and related notions of measures in homogeneous structures. Invariant Keisler measures give a natural notion of "size" on the definable subsets of a first-order structure. We are interested in understanding and possibly classifying the invariant measures on different homogeneous structures. For this talk we focus mainly on homogeneous hypergraphs (and especially 3-hypergraphs).

We classify the invariant Keisler measures on all universal homogeneous k-hypergraphs. This extends results of Albert, who classified all invariant Keisler measures on the random graph. Moreover, our classification relies on proving correspondences between invariant Keisler measures and other notions of measures studied in the context of first-order structures (Ackermann, Freer, & Patel; Crane & Townser; Jahel & Joseph).

Our results on k-hypergraphs imply that all universal homogeneous kay-graphs have a unique invariant Keisler measure. We conclude with a discussion of the measures on the universal homogeneous tetrahedron-free 3-hypergraph.