Abstract
Given a coloring of a set, classical Ramsey theory looks for various configurations within a color class. Rainbow configurations, also called anti-Ramsey configurations, are configurations that occur across distinct color classes. We present some very general results about the types of colorings that will guarantee various types of rainbow configurations in finite settings, as well as several illustrative corollaries. The main goal of this note is to present a flexible framework for decomposing finite sets while guaranteeing the existence of some desired structure across the decomposition.
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Notes
- 1.
For example, if \(s> \log q\), that would suffice, but in fact this works for any slowly growing function in q.
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Acknowledgements
This work was supported in part by NSF Grant DMS 1559911.
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Desgrottes, M., Senger, S., Soukup, D., Zhu, R. (2020). A General Framework for Studying Finite Rainbow Configurations. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory III. CANT 2018. Springer Proceedings in Mathematics & Statistics, vol 297. Springer, Cham. https://doi.org/10.1007/978-3-030-31106-3_5
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