Result 171, Combinatorics

The hypercube Ramsey conjecture

Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is Θ(2n)\Theta(2^n). 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.

Proof

The bigger picture

Why it matters

How large must a red-blue network be before it necessarily contains a one-color hypercube? The manuscript reports that a fixed multiple of the cube's own number of vertices always suffices.

What changes?

The n-dimensional binary cube has 2^n vertices, represented by binary strings of length n; edges join strings differing in exactly one position. Its two-color Ramsey number is the smallest size of a complete graph, with every pair joined, whose every red-blue edge coloring contains a copy with all cube edges in one color. The manuscript claims this number is at most C times 2^n, for an absolute constant C independent of n, resolving the Burr-Erdős hypercube Ramsey conjecture.

What does that help mathematicians do?

At least 2^n vertices are needed simply to fit the cube. Combined with that elementary lower bound, the claimed upper bound pins down the Ramsey number's growth up to constant factors. It rules out any additional factor that grows with dimension. Researchers can therefore deduce that even adversarial edge colorings cannot require a dimension-dependent multiplicative increase in the number of vertices needed to accommodate a monochromatic cube.

Are there practical applications?

The immediate value is foundational: the result identifies the correct growth scale for forcing a highly structured graph inside arbitrary two-color edge patterns. It supplies a sharp order-of-growth benchmark for Ramsey theory. The supplied abstract does not describe an efficient procedure for finding the cube or a practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The hypercube Ramsey number has linear order

September 23, 2026 172 pages

We prove that the two-color Ramsey number of the n-dimensional binary cube is at most C2nC2^n, where C is an absolute constant. This resolves positively the hypercube Ramsey conjecture of Burr and Erdős.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-hypercube-Ramsey-number-has-linear-order-September-23-2026,
  author = {{OpenAI}},
  title = {{The hypercube Ramsey number has linear order}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf}{OAI:The-hypercube-Ramsey-number-has-linear-order-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.