A two-page disproof of the Borsuk partition conjecture

It is presented the simplest known disproof of the Borsuk conjecture stating that if a bounded subset of n-dimensional Euclidean space contains more than n points, then the subset can be partitioned into n+1 nonempty parts of smaller diameter. The argument is due to N. Alon and is a remarkable application of combinatorics and algebra to geometry. This note is purely expository and is accessible for students.

💡 Research Summary

The paper presents a concise, self‑contained exposition of N. Alon’s disproof of the Borsuk partition conjecture. The conjecture, formulated by Karol Borsuk in the 1930s, asserts that any bounded subset of ℝⁿ containing more than n points can be partitioned into n + 1 non‑empty subsets each of strictly smaller diameter. While the statement is true in the plane (Borsuk’s original theorem) and for low dimensions, it was shown to be false for sufficiently large n. The first counterexample was given by Kahn and Kalai (1993) using sophisticated probabilistic and combinatorial arguments. Alon’s construction, however, is remarkably elementary: it relies only on elementary combinatorics, linear algebra over the rationals, and a simple number‑theoretic fact about primes.

The core of the argument is the following. Choose a prime p and set n = 4p. Define the set

 M = { x = (x₁,…,x_n) ∈ {±1}ⁿ : x₁ = 1 and x₂·x₃·…·x_n = 1 }.

Thus each vector in M has first coordinate 1, the remaining coordinates are ±1, and the product of the last n‑1 coordinates equals 1 (equivalently, an even number of –1’s). The cardinality of M is |M| = 2^{n‑2}.

For each x∈M we associate an n×n symmetric matrix

 f(x) =