The vertex size-Ramsey number

In this paper, we study an analogue of size-Ramsey numbers for vertex colorings. For a given number of colors r and a graph G the vertex size-Ramsey number of G, denoted by R̂v(G, r), is the least number of edges in a graph H with the property that any r-coloring of the vertices of H yields a monochromatic copy of G. We observe that Ωr(∆n) = R̂v(G, r) = Or(n ) for any G of order n and maximum degree ∆, and prove that for some graphs these bounds are tight. On the other hand, we show that even 3-regular graphs can have nonlinear vertex size-Ramsey numbers. Finally, we prove that R̂v(T, r) = Or(∆ n) for any tree of order n and maximum degree ∆, which is only off by a factor of ∆ from the best possible.