Definable Kőnig theorems

Abstract

Let X X be a Polish space with Borel probability measure μ , \mu , and let G G be a Borel graph on X X with no odd cycles and maximum degree Δ ( G ) . \Delta (G). We show that the Baire measurable edge chromatic number of G G is at most Δ ( G ) + 1 \Delta (G)+1 , and if G G is μ \mu -hyperfinite then the μ \mu -measurable edge chromatic number obeys the same bound. More generally, we show that G G has Borel edge chromatic number at most Δ ( G ) \Delta (G) plus its asymptotic separation index.