Boolean Relation Theory (BRT) has its origins in the basic Complementation Theorem and Thin Set Theorem. The former asserts that for all strictly dominating f:N^k into N, there exists infinite A contained in N, such that fA = N\A, and the latter asserts that for all f:N^k into N, there exists infinite A contained in N, such that fA is not N. Here fA abbreviates f[A^k], and is viewed as the forward image of f on A.

More generally, BRT investigates statements of the form "for several multivariate functions a certain kind, there are several sets of a certain kind, such that a given Boolean equation (or inequation) holds among these sets and the forward images of the functions on these sets". The truth value of the statement depends on the choice of functions, sets, number of functions, number of sets, and the Boolean equation (inequation).

BRT leads quickly to statements that are provable only by going well beyond the usual ZFC axioms for mathematics. Our forthcoming book focuses on a BRT statement involving two functions and three sets, on N. We survey the results in our forthcoming book coming out in the ASL Lecture Notes in Logic.