Planar point sets with forbidden $4$-point patterns and few distinct distances
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912012720668672 |
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| author | Tao, Terence |
| author_facet | Tao, Terence |
| contents | We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a question of Erdős. The proof combines an analysis by Dumitrescu of forbidden four-point patterns with an algebraic construction of Thiele and Dumitrescu (to eliminate parallelograms), as well as a randomized transformation of that construction (to eliminate most other forbidden patterns). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Planar point sets with forbidden $4$-point patterns and few distinct distances Tao, Terence Combinatorics 52C10, 05B25 We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a question of Erdős. The proof combines an analysis by Dumitrescu of forbidden four-point patterns with an algebraic construction of Thiele and Dumitrescu (to eliminate parallelograms), as well as a randomized transformation of that construction (to eliminate most other forbidden patterns). |
| title | Planar point sets with forbidden $4$-point patterns and few distinct distances |
| topic | Combinatorics 52C10, 05B25 |
| url | https://arxiv.org/abs/2409.01343 |