Descriptive Graph Combinatorics

In this article we survey the emerging field of descriptive graph combinatorics. This area has developed in the last two decades or so at the interface of descriptive set theory and graph theory, and it has interesting connections with other areas such as ergodic theory and probability theory. Our object of study is the theory of definable graphs, usually Borel or analytic graphs on Polish spaces. We investigate how combinatorial concepts, such as colorings and matchings, behave under definability constraints, i.e., when they are required to be definable or perhaps well-behaved in the topological or measure theoretic sense. To illustrate the new phenomena that can arise in the definable context, consider for example colorings of graphs. As usual a Y -coloring of a graph G = (X,G), where X is the set of vertices and G ⊆ X2 the edge relation, is a map c : X → Y such that xGy =⇒ c(x) 6= c(y). An elementary result in graph theory asserts that any acyclic graph admits a 2-coloring (i.e., a coloring as above with |Y | = 2). On the other hand, consider a Borel graph G = (X,G), where X is a Polish space and G is Borel (in X2). A Borel coloring of G is a coloring c : X → Y as above with Y a Polish space and c a Borel map. In contrast to the above basic fact, there are acyclic Borel graphs G which admit no Borel countable coloring (i.e., with |Y | ≤ א0); see Example 3.14. Moreover for each n ≥ 2, one can find acyclic Borel graphs G, which admit a Borel n-coloring but no Borel m-coloring for any m < n; see the first paragraph of Subsection 5,(A). However, sometimes results of classical graph theory have definable counterparts. For example, another standard result in graph theory asserts that every graph of degree at most d admits a (d+ 1)-coloring. It turns out that every Borel graph of degree at most d actually admits a Borel (d+1)-coloring; see Proposition 4.4. Another interesting example of the interplay between combinatorics and definability