E181: “Is the hypercube Ramsey number linear in its vertices?”

Open

Statement	User	Model	Harness	Time
2)V2Every two-colouring of a one-vertex host contains the zero-dimensional cube.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Is there an absolute positive integer C such that every red-blue coloring of the edges of the complete graph…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every two-colouring of a one-vertex host contains the zero-dimensional cube.

kernel-checked, filed Tue Aug 25 2026 07:25:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The zero-dimensional boundary case.

1) V1 Is there an absolute positive integer C such that every red-blue coloring of the edges of the complete graph on C·2^n vertices contains a monochromatic copy of the n-dimensional hypercube, for every n?

open, filed Tue Aug 25 2026 07:25:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk

CubeAdj means Boolean vectors differ in exactly one coordinate. Symmetric Bool functions encode undirected red-blue edge colorings; diagonal values are irrelevant. An injective map preserving one color on cube edges is a monochromatic (not necessarily induced) Q_n copy. A host of exactly C·2^n vertices is equivalent to the O(2^n) Ramsey upper bound by monotonicity.

Scope. All cube dimensions and symmetric two-colourings.

E181:Erdős Problem #181 “Is the hypercube Ramsey number linear in its vertices?”

E181: “Is the hypercube Ramsey number linear in its vertices?”