Monochromatic Polynomial sumset structures on $\mathbb{N}$: an ultrafilter proof
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929313182384128 |
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| author | Goswami, Sayan |
| author_facet | Goswami, Sayan |
| contents | Recently, using machinery's from Ergodic theory, Z. Lian, and R. Xiao proved if $P$ is any polynomial with no constant term, then for every finite coloring of $\mathbb{N}$, there exists two infinite subsets $B,C$ of $\mathbb{N}$ such that the set $\{P(b)+P(c):b\in B, c\in C\}$ is monochromatic. In this article we improve their result by proving that instead of taking such polynomials we can choose any function $f$ having the property that $f(\mathbb{N})\setminus \mathbb{N}$ is finite. We use ultrafilter techniques to prove our result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08724 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monochromatic Polynomial sumset structures on $\mathbb{N}$: an ultrafilter proof Goswami, Sayan Combinatorics 05D10, 54D35 Recently, using machinery's from Ergodic theory, Z. Lian, and R. Xiao proved if $P$ is any polynomial with no constant term, then for every finite coloring of $\mathbb{N}$, there exists two infinite subsets $B,C$ of $\mathbb{N}$ such that the set $\{P(b)+P(c):b\in B, c\in C\}$ is monochromatic. In this article we improve their result by proving that instead of taking such polynomials we can choose any function $f$ having the property that $f(\mathbb{N})\setminus \mathbb{N}$ is finite. We use ultrafilter techniques to prove our result. |
| title | Monochromatic Polynomial sumset structures on $\mathbb{N}$: an ultrafilter proof |
| topic | Combinatorics 05D10, 54D35 |
| url | https://arxiv.org/abs/2404.08724 |