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A note on Ramsey numbers for fans. (English) Zbl 1319.05134

Summary: For two given graphs \(G_{1}\) and \(G_{2}\), the Ramsey number \(R(G_{1},G_{2})\) is the smallest integer \(N\) such that, for any graph \(G\) of order \(N\), either \(G\) contains \(G_{1}\) as a subgraph or the complement of \(G\) contains \(G_{2}\) as a subgraph. A fan \(F_{l}\) is \(l\) triangles sharing exactly one vertex. In this note, it is shown that \(R(F_{n},F_{m})=4n+1\) for \(n\geq \max \{m^{2}-m/2,11m/2-4\}\).

MSC:

05D10 Ramsey theory
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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References:

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