The deviation from right angles in $k$-subsets of points in the plane
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911640354553856 |
|---|---|
| author | Dukes, Peter J. |
| author_facet | Dukes, Peter J. |
| contents | A problem originating with Erdős and Silverman in the 1970s asks for the minimum integer $r(k)$ such that any set of $n \ge r(k)$ points in the plane has some $k$-subset with no right angles. The case $k=4$ has an interesting gap between the known bounds, namely $8 \le r(4) \le 10$. Here, we consider a relaxation that quantifies the deviation from right angles. Specifically, we study $Γ_k(n)$, the supremum of angles $γ$ such that every $n$-set of points in $\mathbb{R}^2$ has a $k$-subset with all angles outside of the interval $90^\circ \pm γ$. We show that $4^\circ \le Γ_4(10) \le 9.292^\circ$. For large $n$, the quantity $Γ_3(n)$ is closely related to a classical minimax angle problem pioneered by Blumenthal, Erdős and Szekeres. We give bounds on $Γ_k(n)$ for a general $k$ and large $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01281 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The deviation from right angles in $k$-subsets of points in the plane Dukes, Peter J. Combinatorics 52C10, 52C35, 05C55 A problem originating with Erdős and Silverman in the 1970s asks for the minimum integer $r(k)$ such that any set of $n \ge r(k)$ points in the plane has some $k$-subset with no right angles. The case $k=4$ has an interesting gap between the known bounds, namely $8 \le r(4) \le 10$. Here, we consider a relaxation that quantifies the deviation from right angles. Specifically, we study $Γ_k(n)$, the supremum of angles $γ$ such that every $n$-set of points in $\mathbb{R}^2$ has a $k$-subset with all angles outside of the interval $90^\circ \pm γ$. We show that $4^\circ \le Γ_4(10) \le 9.292^\circ$. For large $n$, the quantity $Γ_3(n)$ is closely related to a classical minimax angle problem pioneered by Blumenthal, Erdős and Szekeres. We give bounds on $Γ_k(n)$ for a general $k$ and large $n$. |
| title | The deviation from right angles in $k$-subsets of points in the plane |
| topic | Combinatorics 52C10, 52C35, 05C55 |
| url | https://arxiv.org/abs/2605.01281 |