Abstract
We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.
Highlights
The terminology, definitions and some descriptions are taken from two previous works by the first author, namely [1,2].Ramsey numbers and their generalizations have been a fundamentally important area of study in combinatorics for many years
The Ramsey number of two graphs G and H, denoted by r(G, H), is the least t such that any red-blue edge-coloring of Kt contains a red copy of G or a blue copy of H
The Builder can join an endpoint of a red P3 in S1 or S5, which is at the same time the endpoint of a blue path P5, with an endpoint of a blue P5 in the structure obtained after the end of the second r(P4, P5)-game
Summary
The terminology, definitions and some descriptions are taken from two previous works by the first author, namely [1,2].Ramsey numbers and their generalizations have been a fundamentally important area of study in combinatorics for many years. In each round of the game, the Builder chooses an edge and the Painter colors it red or blue. Some new general lower and upper bounds for on-line Ramsey numbers r(C3, Pk) and r(C4, Pk) were proved in [2].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
