Two variants of the size Ramsey number

Given a graph H and an integer r ≥ 2, let G → (H, r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let m(G) = maxF⊆G |E(F )|/|V (F )| and define the Ramsey density minf (H, r) as the infimum of m(G) over all graphs G such that G → (H, r). In the first part of this paper we show that when H is a complete graph Kk on k vertices, then minf (H, r) = (R − 1)/2, where R = R(k; r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál that the size Ramsey number for Kk equals ( R 2 ) . We also study an on-line version of the size Ramsey number, related to the following two-person game: Painter colors on-line the edges provided by Builder, and Painter’s goal is to avoid a monochromatic copy of Kk. The on-line Ramsey number R(k; r) is the smallest number of moves (edges) in which Builder can force Painter to lose if r colors are available. We show that R(3; 2) = 8 and R(k; 2) ≤ 2k(2k−2 k−1 ) , but leave unanswered the question if R(k; 2) = o(R(k; 2)).