On product Schur triples in the integers
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911448661229568 |
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| author | Mattos, Letícia Cecchelli, Domenico Mergoni Parczyk, Olaf |
| author_facet | Mattos, Letícia Cecchelli, Domenico Mergoni Parczyk, Olaf |
| contents | Schur's theorem states that in any $k$-colouring of the set of integers $[n]$ there is a monochromatic solution to $a+b=c$, provided $n$ is sufficiently large. Abbott and Wang studied the size of the largest subset of $[n]$ such that there is a $k$-colouring avoiding a monochromatic $a+b=c$. In other directions, the minimum number of $a+b=c$ in $k$-colourings of $[n]$ and the probability threshold in random subsets of $[n]$ for the property of having a monochromatic $a+b=c$ in any $k$-colouring were investigated. In this paper, we study natural generalisations of these streams to products $ab=c$, in a deterministic, random, and randomly perturbed environments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_18796 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On product Schur triples in the integers Mattos, Letícia Cecchelli, Domenico Mergoni Parczyk, Olaf Combinatorics Schur's theorem states that in any $k$-colouring of the set of integers $[n]$ there is a monochromatic solution to $a+b=c$, provided $n$ is sufficiently large. Abbott and Wang studied the size of the largest subset of $[n]$ such that there is a $k$-colouring avoiding a monochromatic $a+b=c$. In other directions, the minimum number of $a+b=c$ in $k$-colourings of $[n]$ and the probability threshold in random subsets of $[n]$ for the property of having a monochromatic $a+b=c$ in any $k$-colouring were investigated. In this paper, we study natural generalisations of these streams to products $ab=c$, in a deterministic, random, and randomly perturbed environments. |
| title | On product Schur triples in the integers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.18796 |