On Sets of Points in General Position That Lie on a Cubic Curve in the Plane
| Autor: | Mehdi Makhul, Rom Pinchasi |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Studia Scientiarum Mathematicarum Hungarica. 59:196-208 |
| ISSN: | 1588-2896 0081-6906 |
| DOI: | 10.1556/012.2022.01527 |
| Popis: | Let P be a set of n points in general position in the plane. Let R be a set of points disjoint from P such that for every x, y € P the line through x and y contains a point in R. We show that if is contained in a cubic curve c in the plane, then P has a special property with respect to the natural group structure on c. That is, P is contained in a coset of a subgroup H of c of cardinality at most |R|.We use the same approach to show a similar result in the case where each of B and G is a set of n points in general position in the plane and every line through a point in B and a point in G passes through a point in R. This provides a partial answer to a problem of Karasev.The bound is best possible at least for part of our results. Our extremal constructions provide a counterexample to an old conjecture attributed to Jamison about point sets that determine few directions. |
| Databáze: | OpenAIRE |
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