Monk algebras and Ramsey theory

Abstract

Part of this paper is a study of a few finite integral relation algebras that arose in a search for examples of small relation algebras that are weakly representable but not representable. We find several non-representable algebras by splitting atoms and varying the cycle structures of some integral relation algebras without 1-cycles. The absence of 1-cycles puts an upper bound on the number of points in any representation. Our proofs of non-representability typically use standard combinatorial facts about colorings of the edges of complete graphs. One of our proofs required a (possibly) new combinatorial fact: the edges of the complete graph on 11 vertices cannot be colored with three colors in such way as to avoid not only monochromatic triples (as in Ramsey theory) but also trichromatic triples. The main result of this paper generalizes this observation. A coloring of the edges of a complete graph is a bicoloring if, for any three vertices, the three edges connecting them are colored with exactly two colors (neither monochromatic nor trichromatic). Let β(n) be the largest number such that the complete graph on β(n) vertices can be bicolored with n colors. Then β(n) is 5n if n is even, and 2⋅5n−12 if n is odd. Any coloring of the edges of a complete graph with any more vertices must have a monochromatic or trichromatic triple. One of our proofs of non-representability depends on β(3)=10.