Borsuk conjectured in 1933 that any bounded subset of *n*-dimensional Euclidean space can be partitioned into *n*+1 parts, all of smaller diameter. An equilateral simplex shows that that you need at least *n*+1 parts. The famous Borsuk-Ulam theorem can be formulated as saying that you also need at least *n*+1 parts for the Euclidean (*n*-1)-sphere. (Note that *n*+1 parts also suffice for the sphere.)

Borsuk’s conjecture is true in dimensions 2 and 3. In 1993 Kahn and Kalai proved that the conjecture is false if the dimension is sufficiently large. For a short explanation by Gil Kalai, click here. For a wonderful (very) short story on how their proof came about, click here. The lowest dimension for which it is currently known that the conjecture fails is *n*=298, due to Aicke Hinrichs and Christian Richter.

It is not so difficult to prove the 2-dimensional case. The 3-dimensional case, first proved by Eggleston, is somewhat messy. A short proof for finite sets in 3-space was found by Heppes and Revesz (1956). It goes as follows.

**The number of diameter pairs in a set ***S* of *n* points in 3-space is at most 2*n*-2. (A diameter pair of *S* is a pair of points in *S* whose distance equals the diameter of *S*.)

**Therefore, the diameter graph of ***S* has average degree at most 4-4/*n*. So we can always find a point incident to at most 3 diameters of *S*.

**So we can greedily 4-colour the graph.**

**The colour classes give the partition!**

**Problem**: Can this argument be modified easily to also work for infinite sets? For example, a solution to the following problem would imply Eggleston’s theorem:

*Show that there exist such that the graph G of almost-diameter pairs on any finite set S has chromatic number at most 4.*

Here we define *G* of almost-diameter pairs to be the graph on *S* with edges all pairs of points at distance at least times the diameter of *S*.

In general, it is not clear how to go from finite sets to general bounded sets. So it is “conceivable” for example that the conjecture holds for all finite sets in 4-dimensional space, but not for all bounded sets….

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Dear Konard,

I think it is a very good question if the bounds for Borsuk’s problem are the same for finite sets and infinite sets. I vaguely remember (but I am not sure) that the proof for the finite case in space did lead to a simpler proof for the entire problem.

Dear Gil,

I would love to see how to prove the general case in 3-space from the finite case. I have no idea how!