Remark: Family 184. Coloring and independence in graphs with forbidden subgraphs

Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph $F$ need $O_F(\Delta /\log \Delta )$ colors when their maximum degree $\Delta$ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed $r\ge 4$, every $n$-vertex $K_r$-free graph of average degree $d\ge 2$ has an independent set of size ${\Omega }_r(n\log d/d)$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.