OpenAI research collection · 1 manuscript · Source checked 2026-10-19
Colour every connection in a sufficiently large complete network red or blue. The paper claims that a copy of a high-dimensional cube must appear with all its connections the same colour, using only a constant multiple of the cube's number of vertices.
This is a claim about the amount of structure forced by size alone. It does not say the entire network becomes orderly.
Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is Θ(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.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
We prove that the two-color Ramsey number of the n-dimensional binary cube is at most C2^n, where C is an absolute constant. This resolves positively the hypercube Ramsey conjecture of Burr and Erdős.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
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