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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 content is Section 8, "A 63-dimensional almost-counterexample", entirely on p. 4. See the source card. The notation $G$, $V$, $y_i$, $B_1,B_2,B_3$, $C$ and $p$ is that of section_7.

Statement

The $320$ vectors $\{y_i : i\in C\}$ span a space of dimension at most $63$. The vector $q$ in $\mathbb R^{416}$ equal to $2$ on $B_1$, $-1$ on $B_2\cup B_3$ and $0$ elsewhere is orthogonal to $p$ and to every $y_i$ with $i\in C$, but not to every $y_i$ with $i\in C\cup B_1$, so the dimension drops by at least one from the bound $64$ of Section 7.

The paper further states, without proof, that $C$ can be divided into $64$ five-cliques, so that $\{y_i:i\in C\}$ divides into $64$ parts of smaller diameter. It reports that a computation found exactly one such partition in which, for each of the $64$ five-cliques, the isotropic-point sets of its five vertices have a common intersection of size $3$.

Qualifications printed in the section (p. 4).

• As in Section 7, the paper says the dimension bounds stay valid with equality; no proof of the equalities is given.
• The partition into $64$ five-cliques is stated with its proof not included, and the uniqueness of the partition with the extra property is reported as a computational check.

Since $64=63+1$, this set is not a counterexample in dimension $63$: the section's title calls it an almost-counterexample, and the counting bound of five vectors per part needs more than $5\cdot64=320$ points to force more than $64$ parts.

Proof pointer and dependencies

The dimension bound is the inner-product computation on p. 4, which uses the neighbour counts of Section 6 (p. 3), checked by the program G24CHK and taken as given; the five-clique partition and its uniqueness are reported without proof. Read depth: claims checked, on the page images of pp. 2--4; the partition was not reconstructed and the program was not run.

Bears on

• E0505: no bearing on the problem's answer; the section records the 320-point core in dimension at most 63 that the public dimension-63 claims compared in borsuk_dimension_63_public_claims extend by one point.