{"title":"On Sets of Points in General Position That Lie on a Cubic Curve in the Plane","authors":"Mehdi Makhul, R. Pinchasi","doi":"10.1556/012.2022.01527","DOIUrl":null,"url":null,"abstract":"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 \n \n 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|.\n 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.\n The bound \n \n 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.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"62 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studia Scientiarum Mathematicarum Hungarica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1556/012.2022.01527","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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.
设P是平面上一般位置上n个点的集合。设R是与P不相交的点的集合,使得对于每一个x, y - P,经过x和y的直线在R中包含一个点。我们证明如果包含在平面上的三次曲线c中,那么P对于c上的自然群结构有一个特殊的性质,即P包含在基数不超过|R|的c的子群H的余集中。我们用同样的方法来显示一个类似的结果,在这种情况下,B和G中的每个点都是平面上一般位置上的n个点的集合,每条线都经过B中的一个点,G中的一个点经过r中的一个点。这提供了卡拉塞夫问题的部分答案。这个界至少对我们的部分结果来说是最好的。我们的极值结构提供了一个反例,反驳了Jamison关于点集决定很少方向的老猜想。
期刊介绍:
The journal publishes original research papers on various fields of mathematics, e.g., algebra, algebraic geometry, analysis, combinatorics, dynamical systems, geometry, mathematical logic, mathematical statistics, number theory, probability theory, set theory, statistical physics and topology.