  An ultrafilter $p$ on $\omega$ is said to be discrete if, given any function
$f\colon \omega \to X$ to any completely regular Hausdorff space, there is an
$A \in p$ such that $f(A)$ is discrete. Basic properties of discrete
ultrafilters are studied. Three intermediate classes of spaces $\mathscr R_1
\subset \mathscr R_2 \subset \mathscr R_3$ between the class of $F$-spaces and
the class of van~Douwen's $\beta\omega$-spaces are introduced. It is proved
that no product of infinite compact $\mathscr R_2$-spaces is homogeneous;
moreover, under the assumption $\mathfrak d =\mathfrak c$, no product of
$\beta\omega$-spaces is homogeneous.
