Oleg Pikhurko: Measurable edge-colourings of graphings

Consider a graph G=(X,E) on a standard Borel space X with maximum degree bounded by d whose edge set E is a Borel subset of X². It is known that G admits a Borel edge-colouring with 2d-1 colours (Kechris-Solecki-Todorcevic 1999) but not always with 2d-2 colours (Marks 2013). Suppose additionally that we have a probability measure on X such that G is a graphing (i.e. it can be represented by finitely many measure-preserving maps). Then there is a measurable d+o(d) edge-colouring. This is a joint work with Endre Csoka and Gabor Lippner.