Palette Sparsification via FKNP

A random set $S$ is $p$-spread if, for all sets $T$, $$\mathbb{P}(S\supseteq T)\le p^{\mathrm{\mid }T\mathrm{\mid }}\mathrm{.}$$ There is a constant $C>1$ large enough that for every graph $G$ with maximum degree $D$, there is a $C\mathrm{/}D$-spread distribution on $(D+1)$-colorings of $G$. Making use of a connection between thresholds and spread distributions due to Frankston, Kahn, Narayanan, and Park, a palette sparsification theorem of Assadi, Chen, and Khanna follows.