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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2408.14705 |
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| _version_ | 1866909297174118400 |
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| author | Annor, Dickson Y. B. |
| author_facet | Annor, Dickson Y. B. |
| contents | Let $P$ be a set of $n$ green and $n - k$ red points in $\mathbb{C}^2$. A line determined by $i$ green and $j$ red points such that $i + j \ge 2$ and $|i - j| \le r$ is called \emph{r-equichromatic}. We establish lower bounds for $1$-equichromatic and $2$-equichromatic lines. In particular, we show that if at most $2n-k-2$ points of $P$ are collinear, then the number of $1$-equichromatic lines passing through at most six points is at least $\frac{1}{4}(6n-k(k+3))$, and if at most $\frac{2}{3}(2n - k)$ points of $P$ are collinear, then the number of $2$-equichromatic lines passing through at most four points is at least $\frac{1}{6}(10n - k(k + 5))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14705 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $r$-Equichromatic Lines with few points in $\mathbb{C}^2$ Annor, Dickson Y. B. Combinatorics Let $P$ be a set of $n$ green and $n - k$ red points in $\mathbb{C}^2$. A line determined by $i$ green and $j$ red points such that $i + j \ge 2$ and $|i - j| \le r$ is called \emph{r-equichromatic}. We establish lower bounds for $1$-equichromatic and $2$-equichromatic lines. In particular, we show that if at most $2n-k-2$ points of $P$ are collinear, then the number of $1$-equichromatic lines passing through at most six points is at least $\frac{1}{4}(6n-k(k+3))$, and if at most $\frac{2}{3}(2n - k)$ points of $P$ are collinear, then the number of $2$-equichromatic lines passing through at most four points is at least $\frac{1}{6}(10n - k(k + 5))$. |
| title | On $r$-Equichromatic Lines with few points in $\mathbb{C}^2$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.14705 |