  Let $f(d)$ be the smallest number so that every set in $R^d$ of diameter 1
can be partitioned into $f(d)$ sets of diameter smaller than 1. Borsuk's
conjecture was that $f(d)\! =\!d\!+\!1$. We prove that $f(d)\! \ge\!
(1.2)^{\sqrt d}$ for large~$d$.
