author={Alexander S. Kechris and Slawomir Solecki and Stevo Todorcevic},
  journal={Advances in Mathematics},
We study in this paper graph coloring problems in the context of descriptive set theory. We consider graphs G=(X, R), where the vertex set X is a standard Borel space (i.e., a complete separable metrizable space equipped with its σ-algebra of Borel sets), and the edge relation R ⊆ X^2 is "definable", i.e., Borel, analytic, co-analytic, etc. A Borel n-coloring of such a graph, where 1⩽ n ⩽ N_0 , is a Borel map c: X → Y with card(Y)=n, such that xRy⇒c(x) ≠ {c( y). If such a Borel coloring… 

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Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms.