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October 15, 2025Open Access

Ramsey numbers for sparse graphs versus path or cycle

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If k is odd, the Ramsey number for cycles is 2n-1, demonstrating significant changes in graph theory understanding.
The formula for paths indicates dependence on the independence number and parity of k, showcasing a complex relationship.
Previous bounds for Ramsey numbers depended on much larger vertex counts, indicating a historical breakthrough.
Relaxing edge constraints allows broader applications in graph theory, marking a substantial improvement over decades.

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This paper reveals Ramsey number bounds for sparse graphs in terms of vertices and edges, suggesting new insights.

Fan et al. (2025) studied this question.

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