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Claim. Let $G_n$ be the unit distance graph of $\mathbb R^n$. Frankl and Wilson prove that $\chi(G_n)$ grows exponentially in $n$: the site's remark states their bound as $\chi(G_n)\ge(1+o(1))\,1.2^n$, and Raigorodskii's 2000 note (card) cites it as $(1.207+o(1))^n$. This answers yes the second question of Problem 704. The bound is a geometric consequence of the paper's modular intersection theorem, in the case used here: if $p$ is a prime and a family of $(2p-1)$-subsets of an $n$-set has no two members meeting in exactly $p-1$ elements, then the family has at most $\binom{n}{p-1}$ members. The paper's theorem is more general, bounding uniform families whose pairwise intersections avoid the members' size modulo $p$, under a hypothesis relating the uniformity to $p$ that is not restated here. The $0$-$1$ vectors with $2p-1$ ones, scaled so that two of them are at distance one exactly when the corresponding sets meet in $p-1$ elements, induce a subgraph of $G_n$ whose independent sets are such families, so its chromatic number is at least $\binom{n}{2p-1}/\binom{n}{p-1}$, which is exponential in $n$ for $p$ a fixed fraction of $n$ ($n$ of order $7p$ gives the base $1.207$). The paper is P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), no. 4, 357–368, DOI 10.1007/BF02579457; the publisher's record dates the issue to December 1981 and the page name carries the first day of that month, since the record gives no day. The paper is not held in the library, and the derivation above follows the standard presentation of the bound, not the paper's pages.

Covers. The exponential-growth question: $\chi(G_n)$ grows at least exponentially in $n$. Not covered: the estimate of $\chi(G_n)$ beyond the lower bound, where Raigorodskii raised the base to $1.239\ldots$ and 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 paper is refereed: it appeared in Combinatorica, volume 1 (1981). The site labels the problem OPEN, and its remark (page last edited 10 April 2026) credits Frankl and Wilson with the exponential growth; that remark on an open problem is not an acceptance, so no reviewed evidence is listed.