Borsuk conjecture

The Borsuk conjecture asserted that every bounded subset of ${\mathbb{R}}^d$ with positive diameter can be partitioned into $d+1$ subsets of strictly smaller diameter. It is false in sufficiently high dimension.

Kahn and Kalai represented uniform subsets by quadratic sign vectors so that diameter pairs correspond to one forbidden intersection size. The Frankl-Wilson theorem bounds each smaller-diameter part, forcing exponentially many parts while the ambient dimension grows only quadratically.

Take $N=4p-1$, $|A|=|B|=2p-1$ and sign vectors $u_A=21_A-1$. The Frobenius inner product of two outer products is $(u_A\cdot u_B{)}^2$, proving the formula. The squared Euclidean distance is largest precisely when $|A\cap B|=p-1$, where the inner product is minus one. The common diagonal can be deleted, leaving an isometric representation in dimension $\left(\genfrac{}{}{0ex}{}{N}{2}\right)$ if off-diagonal coordinates are multiplied by $\sqrt{2}$. The Frankl-Wilson theorem then bounds each smaller-diameter class.

Use the balanced cut-vector distance formula with $p=13$. There are $\left(\genfrac{}{}{0ex}{}{51}{25}\right)=247959266474052$ distinct cuts. A subset of strictly smaller diameter contains no pair with side intersection thirteen, so deleting their shared coordinate one and applying the Frankl-Wilson theorem bounds its size by $\left(\genfrac{}{}{0ex}{}{51}{12}\right)=158753389900$. At least 1562 smaller-diameter parts are needed, exceeding $1325+1$. The cut vectors occupy an affine hyperplane in ${\mathbb{R}}^{1326}$, which identifies isometrically with ${\mathbb{R}}^{1325}$ and gives the stated explicit counterexample.

Represent a balanced partition of $[4p]$ by the zero-one vector recording which unordered pairs lie across its cut. Complementary subsets represent the same cut; choose the representative containing coordinate one. Two such cuts differ on the pairs crossing the symmetric difference of their sides, giving the displayed distance. Its maximum is $4p^2$, attained exactly when $t=p$. Every cut has $4p^2$ occupied coordinates, so the vectors lie in an affine hyperplane of dimension at most $\left(\genfrac{}{}{0ex}{}{4p}{2}\right)-1$.

The unit ball in ${\mathbb{R}}^n$ can be covered by at most $9^n$ balls of radius $1/4$. A maximal $1/4$-separated set supplies the centres, and disjoint radius-$1/8$ balls around them fit inside the ball of radius $9/8$.