Many results in finite combinatorics can be extended to infinite structures via compactness - but this transfer is powered by the Axiom of Choice and leads, in general, to highly "pathological" objects. It is natural to ask, which combinatorial constructions can be performed in a "well-behaved" fashion, say, in a Borel or measurable way? This question is addressed in a young branch of descriptive set theory called descriptive combinatorics. We will discuss a class of coloring problems with the requirement that the desired coloring be Baire measurable (i.e., "topologically well-behaved"). The central result of this talk is that the existence of a Baire measurable coloring is equivalent to a purely combinatorial statement, analogs of which have for a long time been studied in finite graph theory with no relation to descriptive set theory.