Fan-complete Ramsey numbers
Fan Chung, Qizhong Lin

TL;DR
This paper investigates the conditions under which certain Ramsey numbers involving fan graphs and complete multipartite graphs are tight, providing new bounds and answering longstanding questions in graph theory.
Contribution
It establishes new lower bounds for fan-graph Ramsey numbers and characterizes when these numbers are tight, improving previous tower-type bounds and addressing Burr's question.
Findings
Proves $K_1 + nK_2$ is $K_p$-good for large $n$
Provides exact formulas for $r(G, K_1 + nH)$ under mild conditions
Improves bounds from tower-type to polynomial in $n$
Abstract
For graphs and , we consider Ramsey numbers with tight lower bounds, namely, where denotes the chromatic number of and denotes the number of vertices in . We say is -good if the equality holds. Let be the join graph obtained from graphs and by adding all edges between the disjoint vertex sets of and . Let denote the union graph of disjoint copies of . We show that is -good if is sufficiently large. In particular, the fan-graph is -good if , improving previous tower-type lower bounds for due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number for the case of , the complete -partite graph with and . In…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
