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For graphs , , , we write if for every red-blue colouring of the edge set of we have a red copy of or a blue copy of in . The size Ramsey number is the minimum number of edges of a graph such that . Erdős and Faudree proved that for the cycle of length and for matchings , the size Ramsey number . We improve their upper bound for and by showing that for and for .
For graphs F, G and H, we write F → (G,H) to mean that any red-blue coloring of the edges of F contains a red copy of G or a blue copy of H. The graph F is Ramsey (G,H)-minimal if F → (G,H) but F* ↛ (G,H) for any proper subgraph F* ⊂ F. We present an infinite family of Ramsey -minimal graphs of any diameter ≥ 4.
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