ONLINE RAMSEY NUMBERS: LONG VERSUS SHORT CYCLES

An online Ramsey game is played between Builder and Painter on an infinite board KBbbN. In every round Builder selects an edge, then Painter colors it red or blue. Both know target graphs H1 and H2. Builder aims to create either a red copy of H1 or a blue copy of H2 in KBbbN as soon as possible, and Painter tries to prevent it. The online Ramsey number r~(H1, H2) is the minimum number of rounds such that the Builder wins. We study r~(Ck, Cn), where k is fixed and n is large. We show that r~(Ck, Cn) = 2n + scrO(k) if k is even, while r~(Ck, Cn) leq 3n + o(n) if k is odd. © by SIAM.