Big Ramsey degrees in universal inverse limit structures

We build a collection of topological Ramsey spaces of trees giving rise to universal inverse limit structures, extending Zheng's work for the profinite graph to the setting of Fra\"{\i}ss\'{e} classes of finite ordered binary relational structures with the Ramsey property. This work is based on the Halpern-L\"{a}uchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that for each such Fra\"{\i}ss\'{e} class, its universal inverse limit structures has finite big Ramsey degrees under finite Baire-measurable colourings. For finite ordered graphs, finite ordered $k$-clique free graphs ($k\geq 3$), finite ordered oriented graphs, and finite ordered tournaments, we characterize the exact big Ramsey degrees.Comment: 20 pages, 5 figure

Big Ramsey degrees using parameter spaces

We show that the universal homogeneous partial order has finite big Ramsey degrees and discuss several corollaries. Our proof uses parameter spaces and the Carlson-Simpson theorem rather than (a strengthening of) the Halpern-L\"auchli theorem and the Milliken tree theorem, which are the primary tools used to give bounds on big Ramsey degrees elsewhere (originating from work of Laver and Milliken). This new technique has many additional applications. To demonstrate this, we show that the homogeneous universal triangle-free graph has finite big Ramsey degrees, thus giving a short proof of a recent result of Dobrinen.Comment: 19 pages, 2 figure