Remark: Family 106. Hardness of coloring three-colorable graphs

It is NP-hard to color a three-colorable graph using any fixed number $c\ge 3$ of colors. More strongly, for every fixed $0<\delta <1/3$, a deterministic polynomial-time reduction from $3$SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size $\delta n$ otherwise, where $n$ is the number of vertices.

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.