Short permutation pattern waves: The complete length-five classification

Abstract

For a permutation pi in Sk, a pi-wave is an increasing sequence of integers whose consecutive gaps have relative order pi. Let g(pi, n) be the maximum cardinality of a pi-wave-free subset of [n], and let P(pi, r) be the least integer N such that every r-coloring of [N] contains a monochromatic pi-wave. We determine the asymptotic orders of both functions for every permutation of length at most five. Existing upper bounds and weak-wave constructions provide sharp bounds for all but eight length-five patterns in S_5. For these eight exceptional alternating patterns, we establish g(pi, n) = Theta((log n)squared) and P(pi, r) = Theta(r(log r)squared).