MONOCHROMATIC VS MULTICOLORED PATHS

Let l and k be positive integers, and let X = {0,1,...,l(k-1)}. Is it true that for every coloring delta: X x X --> {0,1,...} there either exist elements x0 < x1 <...< x(l) of X with delta(x0, x1) = delta(X1,X2)=...=delta(X(l-1),x(l)), or else there exist elements y0 < y1 <...< y(k) of X with delta(y(i-1) y(i)) not-equal delta(y(j-1),y(j)) for all 1 less-than-or-equal-to i < j less-than-or-equal-to k? We prove here that this is the case if either l less-than-or-equal-to 2, or k less-than-or-equal-to 4, or 1 greater-than-or-equal-to (3k)2k. The general question remains open.