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Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "listchromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved theconjecture of Ohba that $\text{ch}(G)=\chi(G)$ when $|V(G)|\le 2\chi(G)+1$. Weextend this to a general upper bound: $\text{ch}(G)\le\max\{\chi(G),\lceil({|V(G)|+\chi(G)-1})/{3}\rceil\}$. Our result is sharp for$|V(G)|\le 3\chi(G)$ using Ohba's examples, and it improves the best-knownupper bound for $\text{ch}(K_{4,\dots,4})$.

Beyond Ohba's Conjecture: A bound on the choice number of $k$-chromatic graphs with $n$ vertices