Hales–Jewett theorem

In mathematics, the Hales–Jewett theorem is a fundamental combinatorial result of Ramsey theory named after Alfred W. Hales and Robert I. Jewett, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure; it is impossible for such objects to be "completely random".

An informal geometric statement of the theorem is that for any positive integers n and c there is a number H such that if the cells of a H-dimensional n×n×n×...×n cube are colored with c colors, there must be one row, column, or certain diagonal (more details below) of length n all of whose cells are the same color. In other words, assuming n and c are fixed, the higher-dimensional, multi-player, n-in-a-row generalization of a game of tic-tac-toe with c players cannot end in a draw, no matter how large n is, no matter how many people c are playing, and no matter which player plays each turn, provided only that it is played on a board of sufficiently high dimension H. By a standard strategy-stealing argument, one can thus conclude that if two players alternate, then the first player has a winning strategy when H is sufficiently large, though no practical algorithm for obtaining this strategy is known.