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We give a short proof of a bound on the list chromatic number of graphs $G$of maximum degree $\Delta$ where each neighbourhood has density at most $d$,namely $\chi_\ell(G) \le (1+o(1)) \frac{\Delta}{\ln \frac{\Delta}{d+1}}$ as$\frac{\Delta}{d+1} \to \infty$. This bound is tight up to an asymptotic factor$2$, which is the best possible barring a breakthrough in Ramsey theory, andstrengthens results due to Vu, and more recently Davies, P., Kang, and Sereni.Our proof relies on the first moment method, and adapts a clever countingargument developed by Rosenfeld in the context of non-repetitive colourings. Asa final touch, we show that our method provides an asymptotically tight lowerbound on the number of colourings of locally sparse graphs.

Colouring locally sparse graphs with the first moment method