A finite basis characterization of α-split colorings

Abstract

Fix t>1, a positive integer, and a=(a 1,…,a t) a vector of nonnegative integers. A t-coloring of the edges of a complete graph is called a -split if there exists a partition of the vertices into t sets V 1,…, V t such that every set of a i +1 vertices in V i contains an edge of color i, for i=1,…, t. We combine a theorem of Deza with Ramsey's theorem to prove that, for any fixed a , the family of a -split colorings is characterized by a finite list of forbidden induced subcolorings. A similar hypergraph version follows from our proofs. These results generalize previous work by Kézdy et al. (J. Combin. Theory Ser. A 73(2) (1996) 353) and Gyárfás (J. Combin. Theory Ser. A 81(2) (1998) 255). We also consider other notions of splitting.