Online Ramsey Games for more than two colors

TL;DR

This paper analyzes an online graph coloring game with multiple colors, establishing tight bounds on game duration for certain graph families, and confirms a conjecture in the field.

Contribution

It provides the first tight bounds for online graph avoidance games with more than two colors using advanced combinatorial techniques.

Findings

Established tight upper bounds for game duration with multiple colors.

Confirmed a conjecture by Marciniszyn, Sp"ohel, and Steger.

Extended understanding of online Ramsey-type games to 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.