Ascending waves

A sequence of integers x₁ < x₂ < ··· < x_k is called an ascending wave of length k if x_{i + 1} - x_i ≤ x_{i + 2} - x_{i + 1} for all 1 ≤ i ≤ k - 2. Let f(k) be the smallest positive integer such that any 2-coloring of {1, 2, ..., f(k)} contains a monochromatic ascending wave of length k. Settling a problem of Brown, Erdös, and Freedman we show that there are two positive constants c₁, c₂ such that c₁k³ ≤ f(k) ≤ c₂k³ for all k ≥ 1. Let g(n) be the largest integer k such that any set A ⊆ {1, 2, ..., n} of cardinality |A| ≥ n 2 contains an ascending wave of length k. We show that there are two positive constants c₃ and c₄ such that c₃(log n)² log log n ≤ g(n) ≤ c₄(log n)² for all n ≥ 1.