The Hales-Jewett Theorem states that for any number of colors and any large enough dimension of a combinatorial cube, coloring the cube's cells with that many colors always produces a monochromatic combinatorial line. Named for Alfred Hales and Robert Jewett, it is a central result of Ramsey theory considered by many to be its deepest classical statement, and it implies Van der Waerden's Theorem as a special case.
Facts
StatementFor any positive integers n and c there exists a positive integer H, depending on n and c, such that for any partition of the H-dimensional combinatorial cube over an n-letter alphabet into c parts, at least one part contains an entire combinatorial line. 1 Classification
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Sources
1. Hales-Jewett Theorem (Wikipedia)
Wikimedia FoundationFormal statement sectionQuote, Formal statement section
The Hales-Jewett theorem then states that for given positive integers n and c, there exists a positive integer H, depending on n and c, such that for any partition of WHn into c parts, there is at least one part that contains an entire combinatorial line.
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