Actually, it appears there is a flaw with the below proof. The spectral bound on the chromatic number assumes a measurable coloring, as we want to say the following:

but this assumes that each is Lesbegue integrable, which in turn requires measurable coloring classes.

## Flaw

Actually, it appears there is a flaw with the below proof. The spectral bound on the chromatic number assumes a measurable coloring, as we want to say the following:

but this assumes that each is Lesbegue integrable, which in turn requires measurable coloring classes.