  We prove a strengthened sector lemma for irreducible, finite-dimensional,
locally finite, essential, cocompact CAT(0) cube complexes under the additional
hypothesis that the complex is \emph{hyperplane-essential}; we prove that every
quarterspace contains a halfspace. In aid of this, we present simplified proofs
of known results about loxodromic isometries of the contact graph, avoiding the
use of disc diagrams.
  This paper has an expository element; in particular, we collect results about
cube complexes proved by combining Ramsey's theorem and Dilworth's theorem. We
illustrate the use of these tricks with a discussion of the Tits alternative
for cubical groups, and ask some questions about "quantifying" statements
related to rank-rigidity and the Tits alternative.

Group Theory;Combinatorics