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Claim. Let $G_n$ be the unit distance graph of $\mathbb R^n$. Raigorodskii proves $\chi(G_n)\ge(\gamma+o(1))^n$ with $\gamma=1.239\ldots$, an explicit constant given by an entropy-type product formula in the solution $(x_0,y_0)=(0.36063\ldots,\,0.063907\ldots)$ of a pair of nonlinear equations, improving the base $1.207$ of [[problems/discrete_geometry/E0704/claims/1981_12_01_frankl_wilson|Frankl and Wilson's bound]]. This answers yes the second question of Problem 704. The proof follows the linear-algebra method: the vertices are the vectors in $\{0,1,-1\}^n$ with a prescribed number of nonzero coordinates and a prescribed number of coordinates equal to $-1$, whose convex hull is a cross-polytope rather than the $0$-$1$ cube of the earlier argument; to each vertex $x$ a polynomial over $\mathbb Z/p\mathbb Z$ vanishing on every vertex that is neither $x$ nor at the critical distance from $x$ is attached, the polynomials are reduced by $x_i^3=x_i$, and the number of linearly independent reduced polynomials is bounded by an explicit double binomial sum $D$, so that $\chi(G_n)\ge M/D$ for $M$ the number of vertices; optimizing the parameters gives the base $\gamma$. The note remarks that $p$ may be a prime power and that further gains by this method need a sharper count. The statement and method are recorded on the library card raigorodskii_2000_chromatic_number_space. The note is A. M. Raigorodskii, On the chromatic number of a space, Uspekhi Mat. Nauk 55 (2000), no. 2, 147–148, DOI 10.4213/rm281, translated as Russian Math. Surveys 55 (2000), no. 2, 351–352. The publisher's record of the translation dates its issue to 30 April 2000 and the record of the original gives the year only, so the page name carries that date.

Covers. The exponential-growth question: $\chi(G_n)$ grows at least exponentially in $n$, with base at least $1.239\ldots$. Not covered: the estimate of $\chi(G_n)$ beyond this lower bound, where Larman and Rogers give the upper bound $(3+o(1))^n$, and the existence of $\lim\chi(G_n)^{1/n}$, which remains open.

Depends on. No page of this wiki.

Acceptance. The note is refereed: it appeared in the journal Uspekhi Matematicheskikh Nauk, volume 55 (2000), with an English translation in Russian Mathematical Surveys. The site labels the problem OPEN, and its remark (page last edited 10 April 2026) credits Raigorodskii [Ra00] with the larger base; that remark on an open problem is not an acceptance, so no reviewed evidence is listed.