  We study the extension of the Kechris-Solecki-Todorcevic dichotomy on
analytic graphs to dimensions higher than 2. We prove that the extension is
possible in any dimension, finite or infinite. The original proof works in the
case of the finite dimension. We first prove that the natural extension does
not work in the case of the infinite dimension, for the notion of continuous
homomorphism used in the original theorem. Then we solve the problem in the
case of the infinite dimension. Finally, we prove that the natural extension
works in the case of the infinite dimension, but for the notion of
Baire-measurable homomorphism.

Logic;Category Theory;General Topology