COLORING NUMBER AND ON-LINE RAMSEY THEORY FOR GRAPHS AND HYPERGRAPHS

Let c, s, t be positive integers. The (c, s, t)-Ramsey game is played by Builder and Painter. Play begins with an s-uniform hypergraph G(0) = (V, E(0)), where E(0)=0 and V is determined by Builder. On the ith round Builder constructs a new edge e(i) (distinct from previous edges) and sets G(i) = (V, E(i)), where E(i) = E(i-1)boolean OR{e(i)}. Painter responds by coloring e(i) with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K(s)(t), the complete s-uniform hypergraph on t vertices; otherwise Painter wins when she has colored all possible edges. We extend the definition of coloring number to hypergraphs so that chi(G) <= col(G) for any hypergraph G and then show that Builder can win (c, s, t)-Ramsey game while building a hypergraph with coloring number at most col(K(s)(t)). An important step in the proof is the analysis of an auxiliary survival game played by Presenter and Chooser. The (p, s, t)-survival game begins with an s-uniform hypergraph H(0) = (V,0) with an arbitrary finite number of vertices and no edges. Let H(i-1) = (V(i-1), E(i-1)) be the hypergraph constructed in the first i-1 rounds. On the i-th round Presenter plays by presenting a p-subset P(i) subset of V(i-1) and Chooser responds by choosing an s-subset X(i) subset of P(i). The vertices in P(i)-X(i) are discarded and the edge X(i) added to E(i-1) to form E(i). Presenter wins the survival game if H(i) contains a copy of K(s)(t) for some i. We show that for positive integers p, s, t with s <= p, Presenter has a winning strategy.