Abstract
We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples (x, y, x + ay) whose entries are from the set {1, …, n}, subject to a coloring with two different colors. Previously, only asymptotic formulas for such bounds were known, and only for \(a\in \mathbb {N}\). Using symbolic computation techniques, these results are extended here to arbitrary \(a\in \mathbb {R}\). Furthermore, we give exact formulas for the minimum number of monochromatic Schur triples for a = 1, 2, 3, 4, and briefly discuss the case 0 < a < 1.
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Acknowledgements
We would like to thank our colleagues Thotsaporn Thanatipanonda, Thibaut Verron, Herwig Hauser, and Carsten Schneider for inspiring discussions, comments, and encouragement. The first author was supported by the Austrian Science Fund (FWF): P29467-N32. The second author was supported by the Austrian Science Fund (FWF): F5011-N15.
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Koutschan, C., Wong, E. (2020). Exact Lower Bounds for Monochromatic Schur Triples and Generalizations. In: Pillwein, V., Schneider, C. (eds) Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-44559-1_13
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DOI: https://doi.org/10.1007/978-3-030-44559-1_13
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