  In combinatorial problems it is sometimes possible to define a
$G$-equivariant mapping from a space $X$ of configurations of a system to a
Euclidean space $\mathbb{R}^m$ for which a coincidence of the image of this
mapping with an arrangement $\mathcal{A}$ of linear subspaces insures a desired
set of linear conditions on a configuration. Borsuk-Ulam type theorems give
conditions under which no $G$-equivariant mapping of $X$ to the complement of
the arrangement exist. In this paper, precise conditions are presented which
lead to such theorems through a spectral sequence argument. We introduce a blow
up of an arrangement whose complement has particularly nice cohomology making
such arguments possible. Examples are presented that show that these conditions
are best possible.
