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on Smallest dimension in which Borsuk’s conjecture fails ($C_{28}$)

The statement

For a bounded set $X\subset \mathbb{R}^n$ , its diameter is

\mathrm{diam}(X)\ :=\ \sup\{\|x-y\|_2:\ x,y\in X\}.

Let $b(X)$ be the smallest integer $m$ such that $X$ can be written as a union

X = X_1 \cup \cdots \cup X_m

with

\mathrm{diam}(X_i) < \mathrm{diam}(X)\qquad \text{for all } i=1,\dots,m.

[WX2022-diam-bX]

Define the Borsuk number in dimension $n$ by

b(n)\ :=\ \sup\{b(X):\ X\subset\mathbb{R}^n \text{ bounded with } |X|\ge 2\}.

[Bon2014-bn]

Borsuk’s partition conjecture (1933) asserts that

b(n)\ \le\ n+1 \qquad \text{for all } n\ge 1.

Equivalently, every bounded set in $\mathbb{R}^n$ can be partitioned into $n+1$ subsets of strictly smaller diameter. [KK1993-borsuk-conj]

We define $C_{28}$ to be the smallest integer $n\ge 1$ such that Borsuk’s conjecture fails in $\mathbb{R}^n$ , i.e.

C_{28}\ :=\ \min\{n\ge 1:\ b(n) > n+1\}.

If Borsuk’s conjecture were true in all dimensions, we would set $C_{28}=\infty$ . Since counterexamples are known, $C_{28}$ is finite but its exact value is unknown. [WX2022-open-4-63] [JB2014-ub-64]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.