  We give a short survey of problems and results on (1) diameter graphs and
hypergraphs, and (2) geometric Ramsey theory. We also make some modest
contributions to both areas. Extending a well known theorem of Kahn and Kalai
which disproved Borsuk's conjecture, we show that for any integer $r\ge 2$,
there exist $\varepsilon=\varepsilon(r)>0$ and $d_0=d_0(r)$ with the following
property. For every $d\ge d_0$, there is a finite point set
$P\subset\mathbb{R}^d$ of diameter $1$ such that no matter how we color the
elements of $P$ with fewer than $(1+\varepsilon)^{\sqrt{d}}$ colors, we can
always find $r$ points of the same color, any two of which are at distance $1$.
