An Unsure Note on an Un-Schur Problem
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908994515238912 |
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| author | Parczyk, Olaf Spiegel, Christoph |
| author_facet | Parczyk, Olaf Spiegel, Christoph |
| contents | Graham, Rödl, and Ruciński originally posed the problem of determining the minimum number of monochromatic Schur triples that must appear in any 2-coloring of the first $n$ integers. This question was subsequently resolved independently by Datskovsky, Schoen, and Robertson and Zeilberger. Here we suggest studying a natural anti-Ramsey variant of this question and establish the first non-trivial bounds by proving that the maximum fraction of Schur triples that can be rainbow in a given $3$-coloring of the first $n$ integers is at least $0.4$ and at most $0.66364$. We conjecture the lower bound to be tight. This question is also motivated by a famous analogous problem in graph theory due to Erdős and Sós regarding the maximum number of rainbow triangles in any $3$-coloring of $K_n$, which was settled by Balogh et al. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22024 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Unsure Note on an Un-Schur Problem Parczyk, Olaf Spiegel, Christoph Combinatorics Number Theory 05D10, 11B75, 05C55, 05C15 Graham, Rödl, and Ruciński originally posed the problem of determining the minimum number of monochromatic Schur triples that must appear in any 2-coloring of the first $n$ integers. This question was subsequently resolved independently by Datskovsky, Schoen, and Robertson and Zeilberger. Here we suggest studying a natural anti-Ramsey variant of this question and establish the first non-trivial bounds by proving that the maximum fraction of Schur triples that can be rainbow in a given $3$-coloring of the first $n$ integers is at least $0.4$ and at most $0.66364$. We conjecture the lower bound to be tight. This question is also motivated by a famous analogous problem in graph theory due to Erdős and Sós regarding the maximum number of rainbow triangles in any $3$-coloring of $K_n$, which was settled by Balogh et al. |
| title | An Unsure Note on an Un-Schur Problem |
| topic | Combinatorics Number Theory 05D10, 11B75, 05C55, 05C15 |
| url | https://arxiv.org/abs/2410.22024 |