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Source. Thomas Jenrich, A 64-dimensional two-distance counterexample to Borsuk's conjecture, arXiv:1308.0206v6 (20 August 2014), 7 pages. The paper numbers no theorems; its result is the content of Section 7, "The 64-dimensional counterexample", which runs from p. 3 to p. 4. See the source card.

Setting

Let $G$ be the $G_2(4)$ graph, a strongly regular graph with parameters $(416,100,36,20)$, on vertex set $V$, and let $A$ be its adjacency matrix (Sections 2--3, pp. 1--2). With smallest eigenvalue $s=-4$, the vectors $y_i$, $i\in V$, are the columns of $A+4I$: each has a $4$ in position $i$, a $1$ in each of the $100$ positions adjacent to $i$, and $0$ elsewhere. For distinct $i,j$, $\|y_i-y_j\|^2$ is $144$ when $i,j$ are adjacent and $192$ otherwise, so the $y_i$ form a two-distance set; since the positive eigenvalue has multiplicity $f=65$, they span a space of dimension at most $65$ (p. 2). A subset of the $y_i$ has smaller diameter than the whole set exactly when the corresponding vertices are pairwise adjacent, and, citing Bondarenko, $G$ has no clique of more than $5$ vertices (p. 2).

Section 5 (pp. 2--3) numbers $1,\dots,65$ the $65$ isotropic points of the nondegenerate Hermitian form on $\mathrm{PG}(2,16)$ used in the construction of $G$ (Section 4, p. 2), attaches to each vertex the set of its $15$ isotropic points, lets $B$ be the vertices whose set contains the point $1$ and $C=V\setminus B$, and splits $B$ into the vertex sets $B_h$ of the connected pieces of the subgraph induced on $B$. The program G24CHK checks (Section 6, p. 3) that there are three pieces $B_1,B_2,B_3$ of $32$ vertices each, so $|B|=96$ and $|C|=320$, and that each $i\in V$ has exactly $20$ neighbours in $B_h$ when $i\in B_h$, none when $i\in B\setminus B_h$, and $8$ when $i\in C$. Sections 7 and 8 take these graph facts as given.

Statement

The $352$ vectors $\{y_i : i\in C\cup B_1\}$ form a two-distance set spanning a space of dimension at most $64$, and any subset of them of smaller diameter contains at most $5$ vectors. Hence they cannot be divided into fewer than $71$ parts of smaller diameter, and since $71>65$ the paper concludes (p. 4): "Because $71 > 64 + 1$, the answer to Borsuk's question for $n = 64$ is negative."

The dimension bound comes from a vector $p$ in $\mathbb R^{416}$, equal to $1$ on $B_2$, $-1$ on $B_3$ and $0$ elsewhere, which is orthogonal to every $y_i$ with $i\in C\cup B_1$ but not to every $y_i$, $i\in V$; so the dimension drops by at least one from the bound $65$ (pp. 3--4).

Qualifications printed in the section (p. 4).

• The paper says it can be shown, for instance by vector calculations, that the inequalities $\dim\{y_i:i\in V\}\le65$ and $\dim\{y_i:i\in C\cup B_1\}\le64$ hold with equality, and that the proofs are not included. The counterexample needs only the upper bounds.
• It reports Bondarenko's remark that a computer check had shown at least $72$ parts are needed; the section itself proves $71$.

Proof pointer and dependencies

The argument is the inner-product computation of Section 7 (pp. 3--4), which uses the neighbour counts of Section 6 to evaluate $\langle p,y_i\rangle$.

• The neighbour counts in $B_1,B_2,B_3$ and the sizes $|B_h|=32$ rest on the program G24CHK, distributed with the arXiv source; the paper notes that only the count of $8$ neighbours for $i\in C$ needs the actual construction (p. 3). The program has not been run for this page.
• The clique bound is taken from Bondarenko's arXiv paper on two-distance sets, cited as [2] (p. 2); the bound $65$ follows from the eigenvalue multiplicity $f=65$ (p. 2); the remark on $72$ parts is cited from the published version, [3] (p. 4).

Read depth: claims checked. Sections 2--8 were read on the page images of pp. 1--4, clause by clause for the statement and its setting; the computational facts and the cited bounds were not re-derived.

The joint paper with Brouwer, which p. 1 says follows the principal idea of this manuscript but avoids the extensive computational part, is recorded at theorem_1; this manuscript is a separate source and not its locator.

Bears on

• E0505: after scaling to diameter one, the section's set gives a negative answer in dimension 64, resting on the computer-checked graph facts of Section 6 and on Bondarenko's clique bound, which the manuscript takes as given.