Hales–Jewett theorem

For every finite alphabet and number of colors, sufficiently high-dimensional words over that alphabet contain a monochromatic combinatorial line under any coloring.

Combinatorial line: A set of words formed by assigning one shared symbol to a fixed nonempty set of coordinate positions while fixing all other positions. The theorem guarantees a line whose words all receive the same color.
Cartesian product: A set of ordered tuples formed by choosing one element from each of several sets. A word space is a Cartesian power of its alphabet.
Alfred W. Hales: An American mathematician who coauthored the Hales–Jewett theorem with Robert I. Jewett. He jointly established the theorem in 1963.
Tic-tac-toe: A two-player game in which players place marks on a grid and seek a completed row, column, or diagonal. The theorem explains why sufficiently high-dimensional tic-tac-toe variants have unavoidable monochromatic winning lines.
Word space: The set of all words of a fixed length over a specified alphabet. Colorings of this space are the theorem’s input.
Finite alphabet: A finite set of symbols from which words are formed. Its size is one of the theorem’s parameters.
Robert I. Jewett: An American mathematician who coauthored the Hales–Jewett theorem with Alfred W. Hales. His work with Hales introduced the theorem.
Arithmetic progression: A sequence of numbers with a constant difference between consecutive terms. Encoding integers as words lets the theorem force monochromatic progressions.
Coloring (combinatorics): An assignment of colors to the elements of a set, often used to expose unavoidable patterns. The theorem holds for every assignment of finitely many colors to the word space.
Combinatorial cube: A structured family of tuples in which selected coordinates vary independently over a common set of symbols. A combinatorial line is the one-dimensional instance of this kind of structure.

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Ramsey's theoremCompared with: It is a finite partition theorem whose line structure yields Ramsey-type consequences.