2019/03/30 by Natasha Dobrinen, Dobrinen, Natasha
Computer Science · Mathematics · #03E02 #03E75 #05D10 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.1904.00266
openalex publication_date 2019/03/30 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
The well-known Galvin-Prikry Theorem states that Borel subsets of the Baire\nspace are Ramsey: Given any Borel subset \X\⊆\n[\ω]\ω, where [\ω]\ω is endowed with the metric\ntopology, each infinite subset X\⊆ \ω contains an infinite subset\nY\⊆ X such that [Y]\ω is either contained in \X or\ndisjoint from \X. Kechris, Pestov, and Todorcevic point out in their\nseminal 2005 paper the dearth of similar results for homogeneous structures.\nSuch results are a necessary step to the larger goal of finding a\ncorrespondence between structures with infinite dimensional Ramsey properties\nand topological dynamics, extending their correspondence between the Ramsey\nproperty and extreme amenability. In this article, we prove an analogue of the\nGalvin-Prikry theorem for the Rado graph. Any such infinite dimensional Ramsey\ntheorem is subject to constraints following from the 2006 work of Laflamme,\nSauer, and Vuksanovic. The proof uses techniques developed for the author's\nwork on the Ramsey theory of the Henson graphs as well as some new methods for\nfusion sequences, used to bypass the lack of a certain amalgamation property\nenjoyed by the Baire space.\n