Monochromatic Vs multicolored paths

Let l and k be positive integers, and let X = {0, 1, . . . , l}. Is it true that for every coloring δ : X × X → {0, 1, . . .} there either exist elements x0 < x1 < . . . < xl of X with δ(x0, x1) = δ(x1, x2) = . . . = δ(xl−1, xl), or else there exist elements y0 < y1 < . . . < yk of X with δ(yi−1, yi) 6= δ(yj−1, yj) for all 1 ≤ i < j ≤ k? We prove here that this is the case if either l ≤ 2, or k ≤ 4, or l ≥ (3k). The general question remains open.