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Ramsey Results for Cycle Spectra

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Abstract

Let inline image denote the set of lengths of cycles of a graph G of order n and let inline image denote the complement of G. We show that if inline image, then inline image contains all odd ℓ with inline image and all even ℓ with inline image, where inline image and inline image denote the maximum odd and the maximum even integer in inline image, respectively. From this we deduce that the set inline image contains at least inline image integers, which is sharp.

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