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.