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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 many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Bore
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 om
We show that a locally finite Borel graph is nonsmooth if and only if it admits marker sequences which are far from every point. Our proof uses the Galvin-Prikry theorem and the Glimm-Effros dichotomy.
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, opera
We settle a question posed by G. Eric Moorhouse on the model theory and existence of locally finite generalized quadrangles. In this paper, we completely handle the case in which the generalized quadrangles have a countable number of elements.
A graph $G$ is said to be ubiquitous, if every graph $Gamma$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every locally finite g