Online and Connected Online Ramsey Numbers of a Matching versus a Path

The (G1,G2)-online Ramsey game is a two-player turn-based between builder and painter. Starting from an empty graph with infinite vertices, the adds new edge in each round, painter colors it red or blue. aims to force either copy of G1 blue G2 as few rounds possible, while painter’s aim opposite. online number r˜(G1,G2) minimum edges that needs win game, regardless strategy. Furthermore, we initiate study connected which identical usual one, except at any time induced by all should be connected. In this paper, show general bound matching versus path determine its exact value when has order three four. For version, obtain numbers path.