# Coloring number and on-line Ramsey theory for graphs and hypergraphs

## Abstract

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 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∪{e i }. Painter responds by coloring e i with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K t s , 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 t s ). 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 t s for some i. We show that for positive integers p,s,t with sp, Presenter has a winning strategy.

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Kierstead, H.A., Konjevod, G. Coloring number and on-line Ramsey theory for graphs and hypergraphs. Combinatorica 29, 49–64 (2009). https://doi.org/10.1007/s00493-009-2264-1