User blog comment:KthulhuHimself/nth-level Graham's numbers/@comment-1605058-20151206130222

First of all, what is called Graham's number is not the answer to combinatorial problem which Graham posed, but rather the enormous upper bound which was provided for it.

Second, you indeed have miswrote the definition. The part after last comma should say "without colouring all the six edges between some four coplanar vertices the same color".

Third, there is a theorem proven by Graham and Rothschild (corollary 7 here) which states: Given integers $$k,l,r$$ there exists an integer $$N(k, I, r)$$ such that if $$n\geq N(k, I, r)$$ and the $$k$$-dimensional subspaces (in original problem $$k=1$$, so we color $$1$$-dimensional subspaces, i.e. line segments) of $$n$$-dimensional hypercube are colored with $$r$$ colors, then there exists an $$l$$-dimensional subspace (originally $$l=2$$, i.e. we consider 4 coplanar points) all of whose $$k$$-subspaces have the same color.

Hence we could consider far more general function by considering the least possible  $$N(k, I, r)$$ in this theorem.