$B$-sets and coloring problems
Sherman K. Stein · Bulletin of the American Mathematical Society · 1970
The set-theoretic concept of ".B-set" was first used by Bernstein in 1908 in dealing with a topological question.Since then it has appeared in number theory, combinatorics, and logic.We consider it in the realm of linear graphs, and in particular, questions of map coloring. Definitions. Let F-{X a :a£:A}be a family of sets.A set B is called a 5-set for this family if Br\X a 5 é 0, all aE^4, and B^X ay all «Gi.A family F need not have a B-set.Observe that if B is a .B-set for F t then so is the complement of B.Consider a map covering 52, the two-dimensional sphere.We shall assume that it is regular, that is, each vertex is of degree three.Each region of the map is a topological cell.Two regions are adjacent if they share at least one edge.A sequence of distinct regions J?i, i?2, • • • » Rn, wè 3, is a cycle of length n if Ri is adjacent to JR*+I, lgigw -1, and R n is adjacent to R\.The cycle is odd or even according as n is odd or even.