Online Ramsey turnaround numbers
N\'ora Alm\'asi, Maria Axenovich

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 vertices using colors, for a fixed graph on at most vertices. The goal of Painter is to force a monochromatic copy of , 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 . For sufficiently large , we consider the smallest number of edges so that Painter can always win after edges have been exposed by Builder. In addition, we define to be the smallest such that Painter can always win on a clique with vertices. We give bounds for both…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
