Balanced online Ramsey games in random graphs
Summary: Consider the following one-player game. Starting with the empty graph on \(n\) vertices, in every step \(r\) new edges are drawn uniformly at random and inserted into the current graph. These edges have to be colored immediately with \(r\) available colors, subject to the restriction that each color is used for exactly one of these edges. The player's goal is to avoid creating a monochromatic copy of some fixed graph \(F\) for as long as possible. We prove explicit threshold functions for the duration of this game for an arbitrary number of colors \(r\) and a large class of graphs \(F\). This extends earlier work for the case \(r=2\) by \textit{M. Marciniszyn}, \textit{D. Mitsche}, and \textit{M. Stojaković} [``Online balanced graph avoidance games'', Eur. J. Comb. 28, No.\,8, 2248--2263 (2007; Zbl 1126.05050)]. We also prove a similar threshold result for the vertex-coloring analogue of this game.
scientific article; zbMATH DE number 5541076