Equitable colourings of Borel graphs

Hajnal and Szemeredi proved that if G is a finite graph with maximum degree., then for every integer k >= Delta + 1, G has a proper colouring with k colours in which every two colour classes differ in size at most by 1; such colourings are called equitable. We obtain an analogue of this result for infinite graphs in the Borel setting. Specifically, we show that if G is an aperiodic Borel graph of finite maximum degree Delta, then for each k >= Delta + 1, G has a Borel proper k-colouring in which every two colour classes are related by an element of the Borel full semigroup of G. In particular, such colourings are equitable with respect to every G-invariant probability measure. We also establish a measurable version of a result of Kostochka and Nakprasit on equitable Delta-colourings of graphs with small average degree. Namely, we prove that if Delta >= 3, G does not contain a clique on Delta + 1 vertices and mu is an atomless G-invariant probability measure such that the average degree of G with respect to mu is at most Delta/5, then G has a mu-equitable Delta-colouring. As steps toward the proof of this result, we establish measurable and list-colouring extensions of a strengthening of Brooks' theorem due to Kostochka and Nakprasit.