A first important result in dimension theory is the fact that the Krull dimension of the ring K[X1, . . . , X`] is equal to ` when K is a field. In the literature this result is always obtained after some preliminary efforts that seem excessive for settling such an intuitive fact. For example, many authors rely on the principal ideal theorem of Krull, whose proof is very tricky. Matsumura [6,ch...

For every number n≥2, let Γn be the hypergraph on Rn of arity four consisting all non-degenerate Euclidean rectangles. It is consistent with ZF+DC set theory that chromatic countable while Γn+1 not.