OpenAI · 184-01 · 2026-10-05 · Combinatorics

Correspondence coloring graphs with a forbidden clique

preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026

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

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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

For every fixed integer r ≥ 4, we prove that every Kr-free graph of sufficiently large maximum degree Δ has correspondence chromatic number $O_{r} (Δ/ lo g Δ)$ . 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.

Preprint source and Lean scope

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A Lean scope note is stored for family 184, but it does not name this preprint. It names 184-02. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.