Abstract:
Van der Waerden's Theorem states that for any \(m\), \(k\), we can find an \(n\) so that no matter how we color \([n]\) with \(k\) colors, we can always find a monochromatic arithmetic progression of length \(m\). I will present the idea behind the proof of this, using many (very colorful) pictures. However, we will see that our bound for \(n\) grows faster than the Ackermann function! (This talk is based on lecture notes from the Memphis-Budapest Summer School in Combinatorics.)