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The well-known Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey: Given any Borel subset $mathcal{X}subseteq [omega]^{omega}$, where $[omega]^{omega}$ is endowed with the metric topology, each infinite subset $Xsubseteq omega$ contains an infinite subset $Ysubseteq X$ such that $[Y]^{omega}$ is either contained in $mathcal{X}$ or disjoint from $mathcal{X}$. Kechris, Pestov, and Todorcevic point out in their seminal 2005 paper the dearth of similar results for homogeneous structures. Such results are a necessary step to the larger goal of finding a correspondence between structures with infinite dimensional Ramsey properties and topological dynamics, extending their correspondence between the Ramsey property and extreme amenability. In this article, we prove an analogue of the Galvin-Prikry theorem for the Rado graph. Any such infinite dimensional Ramsey theorem is subject to constraints following from the 2006 work of Laflamme, Sauer, and Vuksanovic. The proof uses techniques developed for the authors work on the Ramsey theory of the Henson graphs as well as some new methods for fusion sequences, used to bypass the lack of a certain amalgamation property enjoyed by the Baire space.
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to
A classical result by Rado characterises the so-called partition-regular matrices $A$, i.e. those matrices $A$ for which any finite colouring of the positive integers yields a monochromatic solution to the equation $Ax=0$. We study the {sl asymmetric
Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number $mathfrak v = mathfrak v(n)$ such that, if $A_1, ldots, A_n$ are $n$ equivalence relations on a common finite ground set $X$, such that for ea
We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $sgeq 3$) that are invariant with respect to the group of all permutations of t
We study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Millikens theorem on trees.