RAMSEY GAME NUMBERS

Abstract. Let H be a graph and m a natural number. Consider the following Game, which is inspired by Ramsey Theory: Player I (blue) and Player II (red) alternate coloring edges of Km with their color. The first player to obtain a monochromatic H in his color wins. The Ramsey Game Number of H (denoted RG(H)) is the least m such that Player I has a winning strategy. We prove (1) if n ≥ 5 then RG(Pn) = n where Pn is the path on n vertices, (2) if n ≤ 5 then RG(Sn) = 2n−3 where Sn = Kn,1, the star graph on n vertices, and (3) if n ≥ 6 then RG(Sn) ≤ 2n−4. A computer program was used to simulate these Games and obtain these results; however, the final game-playing strategies involved in the proofs are humanly comprehensible. 1