Online Ramsey turnaround numbers

TL;DR

This paper studies a strategic online game involving two players on a colored graph, establishing bounds for the minimum edges needed for a player to guarantee a monochromatic copy of a fixed graph, linking it to various extremal graph theory concepts.

Contribution

It introduces and analyzes the online Ramsey turnaround game, providing bounds and connecting it to key extremal graph theory notions.

Findings

Derived bounds for the functions f(n, H) and f(H).

Established relationships with polychromatic colorings and set-coloring Ramsey numbers.

Linked the game to 2-color Turán numbers and other extremal concepts.

Abstract

The online Ramsey turnaround game is a game between two players, Builder and Painter, on a board of $n$ vertices using $3$ colors, for a fixed graph $H$ on at most $n$ vertices. The goal of Painter is to force a monochromatic copy of $H$, the goal of Builder is to avoid this as long as possible. In each round of the game, Builder exposes one new edge and is allowed to forbid the usage of one color for Painter to color this newly exposed edge, and Painter colors the edge according to this restriction. The game is over as soon as Painter manages to achieve a monochromatic copy of $H$. For sufficiently large $n$, we consider the smallest number $f(n, H)$ of edges so that Painter can always win after $f(n, H)$ edges have been exposed by Builder. In addition, we define $f(H)$ to be the smallest $n$ such that Painter can always win on a clique with $n$ vertices. We give bounds for both functions and show that this problem is closely related to other concepts in extremal graph theory, such as polychromatic colorings, set-coloring Ramsey numbers, chromatic Ramsey numbers, and 2-color Tur\'an numbers.