Monochromatic products in random integer sets
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918288760504320 |
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| author | Lidón, Roger Martínez, Darío Morris, Patrick Ortega, Miquel |
| author_facet | Lidón, Roger Martínez, Darío Morris, Patrick Ortega, Miquel |
| contents | A well-known consequence of Schur's theorem is that for $r\in \mathbb{N}$, if $n$ is sufficiently large, then any $r$-colouring of $[n]$ results in monochromatic $a,b,c\in [n]$ such that $ab=c$. In this paper we are interested in the threshold at which the binomial random set $[n]_p$ almost surely inherits this Ramsey-type property. In particular for $r=2$ colours, we show that this threshold lies between $n^{-1/9-o(1)}$ and $n^{-1/11}$. Whilst analogous questions for solutions to (sets of) linear equations are now well understood, our work suggests that both the behaviour of the thresholds and the proof methods needed to determine them differ substantially in the non-linear setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04916 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monochromatic products in random integer sets Lidón, Roger Martínez, Darío Morris, Patrick Ortega, Miquel Combinatorics Number Theory A well-known consequence of Schur's theorem is that for $r\in \mathbb{N}$, if $n$ is sufficiently large, then any $r$-colouring of $[n]$ results in monochromatic $a,b,c\in [n]$ such that $ab=c$. In this paper we are interested in the threshold at which the binomial random set $[n]_p$ almost surely inherits this Ramsey-type property. In particular for $r=2$ colours, we show that this threshold lies between $n^{-1/9-o(1)}$ and $n^{-1/11}$. Whilst analogous questions for solutions to (sets of) linear equations are now well understood, our work suggests that both the behaviour of the thresholds and the proof methods needed to determine them differ substantially in the non-linear setting. |
| title | Monochromatic products in random integer sets |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2512.04916 |