Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Sets in the Plane with Many Concyclic Subsets. | Sets in the Plane with Many Concyclic Subsets . Jeurissen Mathematical Institute Radboud University Toernooiveld Nijmegen The Netherlands Submitted Jul 23 2003 Accepted Aug 7 2005 Published Aug 30 2005 Mathematics Subject Classification 05B30 51M04 Abstract We study sets of points in the Euclidean plane having property R t s every t-tuple of its points contains a concyclic s-tuple. Typical examples of the kind of theorems we prove are a set with R 19 10 must have all its points on two circles or all its points with the exception of at most 9 are on one circle of a set with R 8 5 and N 28 points at least N 3 points lie on one circle a set of at least 109 points with R 7 4 has R 109 7 . We added some results on the analogous configurations in 3-space. 1 Introduction If all points or all points but one of a set V of points in the Euclidean plane are on a circle then clearly every 5-subset of V contains a concyclic 4-subset. In 2 it was proved that the converse also holds unless VI 6. In 1 other proofs were given and also the following was proved. If every 6-subset of a set V VI 77 of points in the Euclidean plane contains a concyclic 4-subset then all points of V with the exception of at most two are on a circle. The same then must hold if the condition is strengthened to every 7-subset contains a concyclic 5-subset. We shall see below Proposition 6 that then the condition VI 77 can be omitted. More generally we investigate sets satisfying the condition one gets by replacing the pair 7 5 by t s t s 3. It may be noteworthy that the essential point of the proofs in 1 and 2 is that the 2- 7 4 2 design the complementary design of the 2- 7 3 1 design has no realisation in the plane with concyclic quadruples as blocks. This means that there is no configuration of 7 points and 7 circles such that every circle contains 4 of the points and every pair of circles intersect in 2 of the points. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R41 1 2 .
