Monochromatic Vs multicolored paths

Abstract

Letl andk be positive integers, and letX={0,1,...,l ^k−1}. Is it true that for every coloring δ:X×X→{0,1,...} there either exist elementsx ₀<x ₁<...<x _l ofX with δ(x ₀,x ₁)=δ(x ₁,x ₂)=...=δ(x _l−1,x _l), or else there exist elementsy ₀<y ₁<...<y _k ofX with δ(y _i−1,y _i) ∈ δ(y _j−1,y _j) 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.