On Ramsey size-linear graphs and related questions
Domagoj Brada\v{c}, Lior Gishboliner, Benny Sudakov

TL;DR
This paper investigates Ramsey numbers for fixed and large graphs, extending previous work on size-linear graphs, and proves new bounds for specific graph subdivisions and related conjectures.
Contribution
It extends existing results on Ramsey size-linear graphs, proving new bounds for subdivisions of K4 and advancing conjectures on Ramsey numbers for connected graphs.
Findings
For subdivisions of K4 with at least 6 vertices, R(H,F) = O(v(F) + e(F)).
Proved the case k=3 for a conjecture relating Ramsey numbers to graph parameters.
Extended the class of graphs with known size-linear Ramsey bounds.
Abstract
In this paper we prove several results on Ramsey numbers for a fixed graph and a large graph , in particular for . These results extend earlier work of Erd\H{o}s, Faudree, Rousseau and Schelp and of Balister, Schelp and Simonovits on so-called Ramsey size-linear graphs. Among others, we show that if is a subdivision of with at least vertices, then for every graph . We also conjecture that if is a connected graph with , then . The case was proved by Erd\H{o}s, Faudree, Rousseau and Schelp. We prove the case .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
