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 G0=(V,E0), where E0=Ø and V is determined by Builder. On the ith round Builder constructs a new edge ei (distinct from previous edges) and sets Gi=(V,Ei), where Ei=Ei−1∪{ei}. Painter responds by coloring ei with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of Kst, 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 χ(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(Kst). 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 H0 = (V,Ø) with an arbitrary finite number of vertices and no edges. Let Hi−1=(Vi−1,Ei−1) be the hypergraph constructed in the first i − 1 rounds. On the i-th round Presenter plays by presenting a p-subset Pi⊆Vi−1 and Chooser responds by choosing an s-subset Xi⊆Pi. The vertices in Pi − Xi are discarded and the edge Xi added to Ei−1 to form Ei. Presenter wins the survival game if Hi contains a copy of Kst for some i. We show that for positive integers p,s,t with s≤p, Presenter has a winning strategy.