Online Ramsey Games for more than two colors

Abstract

Consider the following one-player game played on an initially empty graph with $n$ vertices. At each stage a randomly selected new edge is added and the player must immediately color the edge with one of $r$ available colors. Her objective is to color as many edges as possible without creating a monochromatic copy of a fixed graph $F$. We use container and sparse regularity techniques to prove a tight upper bound on the typical duration of this game with an arbitrary, but fixed, number of colors for a family of $2$-balanced graphs. The bound confirms a conjecture of Marciniszyn, Sp\"ohel and Steger and yields the first tight result for online graph avoidance games with more than two colors.