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Claim. H. G. Eggleston, Covering a three-dimensional set with sets of smaller diameter, J. London Math. Soc. 30 (1955), 11–24. The paper proves Borsuk's assertion in dimension three: every bounded set of diameter $1$ in $\mathbb R^3$ is the union of at most four sets of diameter $<1$. This is the instance $n=3$ of the question, answered yes. The instance $n=1$ is elementary, the plane case is Borsuk's own, and the question is answered no in general by the full claims recorded on this problem's other pages. The page is dated to the publication year, the record giving no day.

Covers. The instance $n=3$ of the question. The site's commentary records the dimension-three result with this paper as its source, and the formal-conjectures statement file for the problem, at its commit of 2026-10-07 (505.lean), states the assertion for $n\le3$ as a solved variant (erdos_505.small_dim), crediting Borsuk for $n=2$ and Eggleston for $n=3$, with no formal proof attached; that file is a statement and not a formalization link. The file it points to, BorsukConjecture.lean, states the case as borsuk_conjecture.three and credits it jointly to Perkal (Colloq. Math. 2 (1947), 45) and to this paper; Kalai's survey (arXiv:1505.04952) credits Eggleston with the first proof for dimension three, and the problem page records why Perkal's note has no claim page. The second file attaches as its formal proof the Lean development linked above, in a fork of that repository, whose proof file names the result as Eggleston's and proves it by the Gale--Grünbaum--Heppes cover method rather than by Eggleston's argument; this corpus has not built it, so it gives no formalized evidence here. The least dimension in which the assertion fails is not part of the site's question; the dimensions $4$ to $8$ are settled by no source recorded here.

Depends on. No page of this wiki.

Acceptance. The result is refereed: the Journal of the London Mathematical Society published the paper. The site's label, DISPROVED (LEAN), credits Kahn and Kalai and Jenrich and Brouwer with the disproof, so the curator's mention of Eggleston is context and not acceptance evidence for this partial claim.