Multipartite graph - space graph Ramsey numbers

The Ramsey number r(F, G) is the least number N such that in every two-coloring (R, B)=(red, blue) of the edges of KN , either there is a red copy of F or else a blue copy of G . The inequality (1) holds in view of the fact that the edges of the complete graph of order (m-1) (n-1) +s-1 can be given the coloring in which the blue graph, denoted (B), is isomorphic to (m-1)K„_~UKS_~ . Then the red graph, denoted (R), does not contain F since ~R) has chromatic number m but the smallest color class has s-1 vertices . Likewise, (B) does not contain G since no component of (B) has more than n-1 vertices. A natural line of inquiry asks for the determination of those cases for which (1) holds with equality . The classical result of this type is the simple theorem of Chvátal [4], namely r(K,,,, T)=(m-1) (n -1)+ for every tree T of order n . Many other examples of equality in (l) can be found when one assumes that G is sufficiently sparse . Previous results of this type can be found in [2], [3] and {5j. The present paper concerns the case in which Fis a multipartite graph. Our notalion for an m-partite graph with parts of size p,,, p, . . ., p will be K(p,, . . . I p,,,) . In case the parts are all of equal size p, we shall write K(p, . . ., p) . The floor of x (greatest integer --x) and the ceiling of x (least integer -x) will be denoted [xl and [x], respectively. A path which is a subgraph of G is a suspended path of G if each vertex of the path, except for its endvertices, has degree 2 in G . Throughout F and G will have no isolates . ComBINAToRicA 5 (4) (1985') 311-318