The First Nontrivial Hales-Jewett Number is Four

We show that whenever the length four words over a three letter alphabet are two-colored, there must exist a monochromatic combinatorial line. We also provide some computer generated lower bounds for some other Hales-Jewett numbers.

[1]  Saharon Shelah,et al.  Primitive recursive bounds for van der Waerden numbers , 1988 .

[2]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[3]  P. Bahr,et al.  Sampling: Theory and Applications , 2020, Applied and Numerical Harmonic Analysis.

[4]  A. Hales,et al.  Regularity and Positional Games , 1963 .