Jan Hubička

We show that the big Ramsey degree of the Boolean algebra with 3 atoms within the countable atomless Boolean algebra is infinite.

In recent years, there has been much progress in the field of structural Ramsey theory, in particular in the study of big Ramsey degrees. In all known examples of infinite structures with finite big Ramsey degrees, there is in fact a single expansion of the structure, called a big Ramsey structure, which correctly encodes the exact big Ramsey degrees of every finite substructure simultaneously. The first half of the article collects facts about this phenomenon that have appeared in the literature into a single cohesive framework, thus offering a conceptual survey of big Ramsey structures. We present some original results indicating that the standard methods of proving finite big Ramsey degrees automatically yield big Ramsey structures, often with desirable extra properties. The second half of the article is a survey in the more traditional sense, discussing numerous examples from the literature and showing how they fit into our framework. We also present some general results on how big Ramsey degrees are affected by expanding structures with unary functions.

We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization, we show that the Fraïssé limit of each such class admits a big Ramsey structure satisfying the infinite Ramsey theorem, implying that the automorphism group of the Fraïssé limit has a metrizable universal completion flow.

We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin's catalogue. 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 coherent EPPA. Our results are a contribution to Ne\v{s}et\v{r}il's classification programme of Ramsey classes and can be seen as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion 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 or the Ramsey property. The main results have numerous consequences for the automorphism groups of the Fraisse limits of the classes. As corollaries, we prove amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).

We show that the universal homogeneous partial order has finite big Ramsey degrees, and we discuss several corollaries. Our proof relies on parameter spaces and the Carlson--Simpson theorem rather than (a strengthening of) the Halpern--L\"auchli theorem and the Milliken tree theorem, which are typically used to bound big Ramsey degrees in existing literature (originating from work of Laver and Milliken). This new technique has many additional applications. We show that the homogeneous universal triangle-free graph has finite big Ramsey degrees, providing a short proof of a recent result by Dobrinen. Moreover, generalizing indivisibility (vertex partition) result of Nguyen van Th\'e and Sauer, we give upper bound on big Ramsey degrees of metric spaces with finitely many distances. This leads to a new combinatorial argument for the oscillation stability of the Urysohn Sphere.

We discuss digitization, subsequent digital analysis and processing of negatives (and diapositives) made by Finlay, Thames, Dufay, Paget, and similar additive color screen processes. These early color processes (introduced in the 1890s and popular until the 1950s) used a special color screen filter and a monochromatic negative. Due to poor stability of dyes used to produce color screens many of the photographs appear faded; others exist only in the form of (monochromatic) negatives. We discuss the possibility of digitally reconstructing the original color from scans of original negatives or by virtue of infrared imaging of original transparencies (which eliminates the physically coupled color filters) and digitally recreating the original color filter pattern using a new open-source software tool. Photographs taken using additive color screen processes are some of the very earliest color images of our shared cultural heritage. They depict people, places, and events for which there are no other surviving color images. We hope that our new software tool can bring these images back to life.