Editing Van der Waerden's theorem (section)

=== Proof in general case ===
The proof for ''W''(2, 3) depends essentially on proving that ''W''(32, 2) ≤ 33.  We divide the integers {1,...,325} into 65 'blocks', each of which can be colored in 32 different ways, and then show that two blocks of the first 33 must be the same color, and there is a block colored the opposite way.  Similarly, the proof for ''W''(3, 3) depends on proving that

: <math>W(3^{7(2 \cdot 3^7+1)},2) \leq 3^{7(2 \cdot 3^7+1)}+1.</math>

By a double [[mathematical induction|induction]] on the number of colors and the length of the progression, the theorem is proved in general.