Theoretical analysis demonstrates Ramsey goodness in large sparse connected graphs paired with disjoint complete graphs, highlighting universal thresholds across component counts.
Let be a connected graph with vertices and edges and denote the disjoint union of complete graphs . In this paper, by developing a trichotomy for sparse graphs, we show that for given integers and , there exists a positive constant such that if and is large, then is ‐good, that is, the Ramsey number is In particular, the above equality holds for any positive integers , and , provided is large. The case was obtained by Burr, Erdős, Faudree, Rousseau, and Schelp, and the case was established by Luo and Peng.
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Zhang et al. (2026) studied this question.
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