Automorphism groups and Ramsey properties of sparse graphs

We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todor\v{c}evi\'c correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an $\omega$-categorical sparse graph has no $\omega$-categorical expansion with extremely amenable automorphism group

Canonical functions: a proof via topological dynamics

Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument. We moreover present equivalent algebraic characterisations of canonicity

Forbidden cycles in metrically homogeneous graphs

Aranda, Bradley-Williams, Hubi\v{c}ka, Karamanlis, Kompatscher, Kone\v{c}n\'y and Pawliuk recently proved that for every primitive 3-constrained space $\Gamma$ of finite diameter $\delta$ from Cherlin's catalogue of metrically homogeneous graphs there is a finite family $\mathcal F$ of $\{1,2,\ldots, \delta\}$-edge-labelled cycles such that each $\{1,2,\ldots, \delta\}$-edge-labelled graph is a (not necessarily induced) subgraph of $\Gamma$ if and only if it contains no homomorphic images of cycles from $\mathcal F$. This analysis is a key to showing that the ages of metrically homogeneous graphs have Ramsey expansions and the extension property for partial automorphisms. In this paper we give an explicit description of the cycles in families $\mathcal F$. This has further applications, for example, interpreting the graphs as semigroup-valued metric spaces or homogenizations of $\omega$-categorical $\{1,\delta\}$-edge-labelled graphs.Comment: 24 pages, 2 table