Ramsey numbers for multiple copies of sparse graphs
Aurelio Sulser, Milo\v{s} Truji\'c

TL;DR
This paper improves bounds on the Ramsey numbers for multiple copies of sparse graphs, especially those with bounded maximum degree, using advanced absorbing methods to refine long-standing theoretical results.
Contribution
It provides significantly stronger bounds on the number of copies needed for the Ramsey number behavior in sparse graphs, extending prior results with new methods.
Findings
Stronger bounds for Ramsey numbers of multiple sparse graphs.
Applicable to graphs with bounded maximum degree.
Methodology based on an efficient absorbing technique.
Abstract
For a graph and an integer , we let denote the disjoint union of copies of . In 1975, Burr, Erd\H{o}s, and Spencer initiated the study of Ramsey numbers for , one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant such that , provided is sufficiently large. Subsequently, Burr gave an implicit way of computing and noted that this long term behaviour occurs when is triply exponential in . Very recently, Buci\'{c} and Sudakov revived the problem and established an essentially tight bound on by showing follows this behaviour already when the number of copies is just a single exponential. We provide significantly stronger bounds on in case is a sparse graph, most notably of bounded maximum degree. These are…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Complexity and Algorithms in Graphs · Advanced Graph Theory Research
