Correspondence coloring graphs with a forbidden clique

For every fixed integer r ≥ 4, we prove that every K${}_r$-free graph of sufficiently large maximum degree Δ has correspondence chromatic number $O_r(\Delta /\log \Delta )$. This resolves the Alon–Krivelevich–Sudakov coloring conjecture in the stronger correspondence-coloring form. The same bound, with a constant depending on F, holds when any fixed graph F is excluded as an ordinary subgraph. Ordinary and list coloring satisfy the same bounds.