On-line Ramsey Theory

On-line Ramsey Theory

The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph H, keeping the constructed graph in a prescribed class G. The main problem is to recognize the wi...

On-line Ramsey Theory for Bounded Degree Graphs

When graph Ramsey theory is viewed as a game, “Painter” 2-colors the edges of a graph presented by “Builder”. Builder wins if every coloring has a monochromatic copy of a fixed graph G. In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class H. Builder wins the game (G,H) if a monochromatic copy of G can be force...

On-line Ramsey Numbers

Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colours them, in either red or blue, as each appears. Builder’s aim is to force Painter to draw a monochromatic copy of a fixed graph G. The minimum number of edges which Builder must draw, regardless of Painter’s strategy, in order to guarantee that this happens is known as the o...

Ramsey Theory

Ramsey Theory

Ramsey Theory is the study of inevitable substructures in large (usually discrete) objects. For example, consider colouring the edges of the complete graph Kn with two colours. In 1930, Ramsey [13] proved that if n is large enough, then we can find either a red complete subgraph on k vertices or a blue complete subgraph on ` vertices. We write Rpk, `q for the smallest such n. Another famous exa...