
TL;DR
This paper investigates the properties of graphs related to Ramsey numbers, introducing new techniques to determine when certain graphs are 'good' in multi-color Ramsey contexts, and applies these to paths of various lengths.
Contribution
The paper develops a novel method for proving that graphs are 2-good and applies it to paths of lengths 5, 6, and 7, advancing understanding of graph goodness in Ramsey theory.
Findings
Established a new technique for proving 2-goodness of graphs.
Proved that paths P5, P6, and P7 are 2-good.
Extended the class of graphs known to be good in Ramsey theory.
Abstract
Given graphs , the Ramsey number is the smallest integer for which in any coloring of the edges of the complete graph with colors , there is some color with a monochromatic copy of . We call a tuple good if for every -coloring of the edges of an -chromatic graph, there is some color with a monochromatic copy of . We call a graph -good if the -tuple is good, and is good if it is -good for every . Bialostocki and Gy\'arf\'as proved that matchings are good and asked whether every acyclic is good. A natural strategy shows that is -good for and that is good. We develop a new technique for showing that a graph is -good, and we apply it successfully to , , and .
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Taxonomy
TopicsData Mining Algorithms and Applications
