Norine’s antipodal edge-colouring conjecture
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The question is whether for every n≥2, a two-colouring of the edges of the n-cube in which antipodal edges have opposite colours contains a monochromatic path from some vertex to its antipode.
Why it matters and what counts as progress
Why it matters: It asks whether antipodal colour reversal alone forces a monochromatic path across a cube in every dimension.
What counts as progress: Prove the antipodal monochromatic-path assertion for every cube dimension or construct an antipodally opposite-coloured counterexample.
How the problem developed
Sergei Norine Conjectured that every red-blue edge-colouring of a hypercube with antipodal edges assigned opposite colours contains a monochromatic path joining a vertex to its antipode.
- 2020 · Partial result [1]
Zdeněk Dvořák For arbitrary red-blue edge-colourings of Q_n, improved the bound for a path between antipodal vertices to (3/8 + o(1))n colour changes; this did not prove Norine’s antipodal-colouring claim.
- 21 Jul 2026 · Proof [2]
Hehui Wu and Ningyuan Yang; GPT 5.6 Sol Ultra generated the central proof idea, and Codex using GPT 5.6 Sol assisted drafting Proved the conjecture for every hypercube Q_n with n ≥ 2 using a chain-level Borsuk–Ulam obstruction.
AI contribution
A Chain-Level Borsuk--Ulam Obstruction Proof of Norine's Antipodal-Coloring Conjecture
For every n≥2, a two-colouring of the edges of the n-cube in which antipodal edges have opposite colours contains a monochromatic path from some vertex to its antipode.
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GPT-origin idea supports a proof of Norine’s antipodal-colouring conjecture