van der Waerden's theorem

Van der Waerden's theorem states that for any chosen number of colors and any length of a arithmetic progression, there’s a minimum number N such that if you color the numbers from 1 to N with those colors, you must find a monochromatic (single-colored) arithmetic progression of that specified length. In simpler terms, no matter how you color a sufficiently long sequence of numbers, a certain length of evenly spaced numbers all sharing the same color will always appear. This theorem highlights the unavoidable patterning in colored sequences and has foundational importance in combinatorics and Ramsey theory.