Let c, s, t be positive integers. We call an s-uniform a hypergraph an s-graph. The (c, s, t)-Ramsey game is played by Builder and Painter. Play begins with an s-graph G_{0} = (V, E_{0}), where E_{0} = ∅ and V is determined by Builder. On the ith round Builder constructs a new edge e_{i} 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-graph on t vertices; otherwise Painter wins when she has colored all possible edges.

We extend the definition of coloring number to s-graphs in a natural way so that χ(G) ≤ ∗col(G) for any s-graph 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}). This relates to, but does not answer, a question of Kurek, Ruciński and Rödl as to whether Builder can win the (2, 2, t)-Ramsey game while constructing an s-graph with substantially fewer than ∗Ram(t)2 edges. Our main tool is the analysis of the following new game.

Let p be positive integers with s ≤ p. The (p, s, t)-survival game is played by two players, Presenter and Chooser. Play begins with the s-graph H_{0} = (S_{0}, E_{0}) as in the Ramsey game. At the beginning of the ith round the players will have constructed an s-graph H_{i−1} = (S_{i−1}, E_{i−1}). During the ith round they construct H_{i} = (S_{i}, E_{i}) as follows. Presenter plays by presenting a p-subset P_{i} ⊆ S_{i−1}. Chooser responds by choosing an s-set X_{i} ⊆ P_{i}. The remaining vertices in P_{i} − X_{i} are discarded, leaving S_{i} = S_{i−1} − (P_{i} − X_{i}) and

E_{i} = (E_{i−1} ∪ {X_{i}}) − {X_{j} ∈ E_{i−1} : X_{j}S_{i}} .

Presenter wins if H_{i} contains the complete s-uniform hypergraph for some i; otherwise Chooser wins when |S_{i}| < t.

For all positive integers p, s, t with s ≤ p, Presenter has a winning strategy in the (p, s, t)-survival game.

This is joint work with Goran Konjevod.