Using Ramsey theory to measure unavoidable spurious correlations in Big Data

Given a dataset we quantify how many patterns must always exist in the dataset. Formally this is done through the lens of Ramsey theory of graphs, and a quantitative bound known as Goodman's theorem. Combining statistical tools with Ramsey theory of graphs gives a nuanced understanding of how far away a dataset is from random, and what qualifies as a meaningful pattern. This method is applied to a dataset of repeated voters in the 1984 US congress, to quantify how homogeneous a subset of congressional voters is. We also measure how transitive a subset of voters is. Statistical Ramsey theory is also used with global economic trading data to provide evidence that global markets are quite transitive.Comment: 21 page

Ramsey expansions of metrically homogeneous graphs

We discuss the Ramsey property, the existence of a stationary independence relation and the coherent extension property for partial isometries (coherent EPPA) for all classes of metrically homogeneous graphs from Cherlin's catalogue, which is conjectured to include all such structures. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and the coherent EPPA. Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's classification programme of Ramsey classes and as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion (or lift or ordering) property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin's classes. The existence of such a "completion algorithm" then allows us to apply several strong results in the areas that imply EPPA and respectively the Ramsey property. The main results have numerous corollaries on the automorphism groups of the Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor revisio

Completing graphs to metric spaces

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions

Amenability and the Hrushovski property for Fraisse classes of directed graphs

This is joint work separately with Miodrag Sokic and Marcin Sabok. Given a Fraisse class, there are the related combinatorial questions: "Does this class have the Hrushovski property?" and "Is the automorphism group of the Fraisse limit an amenable group?". Building on work of Angel-Kechis-Lyons and Zucker the amenability question has recently been answered for all Fraisse classes of directed graphs, and using a Mackey-type construction the Hrushovski question has recently been answered for the same classes. We will survey the results and the techniques used.Non UBCUnreviewedAuthor affiliation: University of TorontoGraduat

Amenability and Unique Ergodicity of the Automorphism Groups of all Countable Homogeneous Directed Graphs

We establish the amenability, unique ergodicity and nonamenability of various automorphism groups from Cherlin's list of countable homogeneous directed graphs. This marks a complete understanding of the amenability of the automorphism groups from this list, and except for the Semigeneric graph case, marks a complete understanding of the unique ergodicity of these groups. Along the way we establish that a certain product of Fraïssé classes preserves amenability, unique ergodicity and the Hrushovski property. We also establish the unique ergodicity of various other automorphism groups of Fraïssé structures that do not appear on Cherlin's list.Ph.D