Monochromatic Vs multicolored paths

Letl andk be positive integers, and letX={0,1,...,lk−1}. Is it true that for every coloring δ:X×X→{0,1,...} there either exist elementsx0<x1<...<xl ofX with δ(x0,x1)=δ(x1,x2)=...=δ(xl−1,xl), or else there exist elementsy0<y1<...<yk ofX with δ(yi−1,yi) ∈ δ(yj−1,yj) for all 1<-i<j≤k? We prove here that this is the case if eitherl≤2, ork≤4, orl≥(3k)2k. The general question remains open.