r-Bounded k-complete bipartite bihypergraphs and generalized split graphs

Let P and Q be hereditary classes of graphs. Suppose that the stability number α(H) is bounded above for all H∈P, and the clique number ω(H) is bounded above for all H∈Q. An ordered partition A∪B=V(G) is called a Ramseian (P,Q)-partition if G(A)∈P and G(B)∈Q, where G(X) denotes the subgraph of G induced by X. Let R(P,Q) be the set of all graphs which have a Ramseian (P,Q)-partition. It is proved that if both P and Q have finite forbidden induced subgraph (FIS) characterizations then R(P,Q) also has such a characterization. In particular, every class of (α,β)-polar graphs (which are generalizations of split graphs) has a finite FIS-characterization. For the proof we use the following model. Let H0 and H1 be hypergraphs with the same vertex set V. The ordered pair H=(H0,H1) is called a bihypergraph. A bihypergraph H=(H0,H1) is called bipartite if there is an ordered partition V0∪V1=V(H) such that Vi is stable in Hi, i=0,1. If the maximum cardinality of hyperedges in H is at most r and every k-subset of V(H) contains at least one hyperedge then H∈C(k,r). We prove that there exists a finite number of minimal non-bipartite bihypergraphs in C(k,r) (when k and r are fixed).