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arXiv · 2112.10222

Definable Kőnig theorems

Matt Bowen·Felix Weilacher
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Abstract

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

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Matt Bowen, Felix Weilacher. 2021-12-19. Definable Kőnig theorems. https://arxiv.org/abs/2112.10222

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