Q-Ramsey classes of graphs

In this paper, we investigate the problem of ( k , Q ) -Ramsey classes of graphs, which were introduced in [M. Borowiecki, A. Fiedorowicz, On partitions of hereditary properties of graphs, Discuss. Math. Graph Theory 26 (2006) 377–387] as an extension of the well-known notion of a vertex Ramsey class of graphs. The definition is based on a concept of a ( k ; Q ) -colouring of graphs. Let Q be a hereditary graph property and assume Q ⊆ O 2 , where O 2 denotes the class of all bipartite graphs. A ( k ; Q ) -colouring of a graph G is a mapping f from the set of vertices of G to the set { 1 , … , k } of colours, such that for any two distinct colours i and j , the subgraph induced in G by all the edges x y such that f ( x ) = i and f ( y ) = j has the property Q . A graph property P is called a ( k , Q ) -Ramsey class of graphs, if for any graph G ∈ P , there is a graph H ∈ P such that in any ( k ; Q ) -colouring of H we have a monochromatic copy of G . ve that some important graph classes, such as k -degenerate graphs, k -trees or hom-properties, are ( k , Q ) -Ramsey classes of graphs. We also provide sufficient conditions for a graph property to be a ( k , Q ) -Ramsey class.