Monochromatic strictly ascending waves and permutation pattern waves

Abstract

We investigate functions related to monochromatic ascending and descending waves. In particular, we show that the minimal integer M such that every 2-coloring of [1,M] admits a monochromatic n-term strictly ascending wave satisfies M=Θ(n4). Taking a definition from the permutation pattern literature, we generalize strictly monotonic waves to progressions x1,x2,…,xk+1 where x2−x1,x3−x2,…,xk+1−xk is order equivalent to a given permutation of {1,2,…,k}.