A Note on On-Line Ramsey Numbers for Some Paths

Tomasz Dzido and Renata Zakrzewska
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Tomasz Dzido: Institute of Informatics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-309 Gdańsk, Poland
Renata Zakrzewska: Mathematics Teaching and Distance Learning Centre, Gdańsk University of Technology, 80-233 Gdańsk, Poland

Mathematics, 2021, vol. 9, issue 7, 1-6

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 P 4 and H is a path P 10 or P 11 .

Keywords: Ramsey number; on-line Ramsey number; path (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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