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Title: An introduction to Borel chromatic numbers.

The Borel chromatic number – introduced by Kechris, Solecki, and Todorcevic (1999) – generalizes the chromatic number on finite graphs to definable graphs on topological spaces. We will see examples of graphs with chromatic number 2 which cannot be colored in a Borel way with less than 3 colors (or even any finite number of colors). Many interesting examples of Borel graphs also arise from the continuous or Borel action of a finitely generated group on a topological space. I will explain the proof that a Borel graph with bounded degree k can always be colored using k+1 colors in a Borel way. Finally I will discuss the problem of characterizing the Borel graphs with finite Borel chromatic number.