We proved with Bowen and Sabok that hyperfinite one-ended bipartite graphings with non-integral fractional perfect matchings admit a measurable perfect matching. We use this to prove the amenable Lyons-Nazarov theorem, give a new proof of measurable circle squaring and show other combinatorial applications. Tímár used it to find a factor perfect matching between independent Poisson point processes in the euclidean space.

Time permitting I show an example of a d-regular treeing for d>2 without measurable perfect matching, this solves a well-known problem of Kechris and Marks.