×

On 5-isosceles sets in the plane, with linear restriction. (English) Zbl 1149.52015

Summary: A finite planar set is \(k\)-isosceles for \(k\geq 3\) if every \(k\)-point subset of the set contains a point equidistant from two others. We show that an 8-set on a line is 5-isosceles if and only if its adjacent interpoint distances are equal to each other, and no 5-isosceles 9-set has 9 points on a line. We also show that the maximum 5-isosceles set with 8 points on a line contains at most 10 points.

MSC:

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fishburn, P.: Isosceles planar subsets. Discrete Comput. Geom. 19, 391–398 (1998) · Zbl 0938.52013
[2] Xu, C., Ding, R.: More about 4-isosceles planar sets. Discrete Comput. Geom. 31, 655–663 (2004) · Zbl 1068.52024
[3] Xu, C., Ding, R.: About 4-isosceles planar sets. Discrete Comput. Geom. 27, 287–290 (2002) · Zbl 1005.52003
[4] Wei, X., Ding, R.: A 4-isosceles 7-point set with both circle and linear restrictions. Ars Comb. 80, 189–191 (2006) · Zbl 1224.52030
[5] Erdös, P., Fishburn, P.: Maximum planar sets that determine k distances. Discrete Math. 160, 115–125 (1996) · Zbl 0868.52007
[6] Erdös, P., Fishburn, P.: Distinct distances in finite planar sets. Discrete Math. 175, 97–132 (1997) · Zbl 0894.52007
[7] Erdös, P., Fishburn, P.: Duplicated distances in subsets of finite planar sets. Geombinatorics 8, 73–77 (1999) · Zbl 0941.52014
[8] Erdös, P., Fishburn, P.: Minimum planar sets with maximum equidistance counts. Comput. Geom. 6, 1–12 (1996)
[9] Altman, E.: On a problem of P. Erdös. Am. Math. Monthly 70, 148–157 (1963) · Zbl 0189.22904
[10] Sedliačková, Z.: k-isosceles planar set. Slov. J. Geom. Graph. 2 (2005) · Zbl 1135.52304
[11] Balint, V., Kojdjaková, Z.: Answer to one of Fishburns questions. Arch. Math. (Brno) 37, 289–290 (2001) · Zbl 1090.52011
[12] Kojdjaková, Z.: 5-rovnoramenné rovinné množiny. Zb. VII. Veduckej Konf. Košice 49–50 (2002) (in Slovak)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.