Large facing tuples and a strengthened sector lemma
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.