Erdős Problem #799

The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.

Is it true that $\chi_L(G)=o(n)$ for almost all graphs on $n$ vertices?

#799: [ERT80]

graph theory | chromatic number

A problem of Erdős, Rubin and Taylor.

The answer is yes: Alon [Al92] proved that in fact the random graph on $n$ vertices with edge probability $1/2$ has\[\chi_L(G) \ll \frac{\log\log n}{\log n}n\]almost surely. Alon, Krivelevich, and Sudakov [AKS99] improved this to\[\chi_L(G) \asymp \frac{n}{\log n}\]almost surely.

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Additional thanks to: David Penman

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