OpenAI · 171-01 · 2026-09-23 · Combinatorics
The hypercube Ramsey number has linear order
preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026
Family 171. The hypercube Ramsey conjecture.
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
Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the -dimensional cube is . Thus every red-blue coloring of a complete graph on a universal constant times the cube's number of vertices contains a monochromatic copy of the cube.
We prove that the two-color Ramsey number of the n-dimensional binary cube is at most , where C is an absolute constant. This resolves positively the hypercube Ramsey conjecture of Burr and Erdős.
Preprint source and Lean scope
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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.