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 =Ø and V is determined by Builder. On the i th 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 ∪{ 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 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( 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 ,Ø) 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 ⊆ V i −1 and Chooser responds by choosing an s -subset X i ⊆ 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.