A Mazurkiewicz set $M$ is a subset of a plane with the property that each straight line intersects $M$ in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set.
@article{119213,
author = {Marta N. Charatonik and W\l odzimierz J. Charatonik},
title = {On Mazurkiewicz sets},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {41},
year = {2000},
pages = {817-819},
zbl = {1052.54030},
mrnumber = {1800162},
language = {en},
url = {http://dml.mathdoc.fr/item/119213}
}
Charatonik, Marta N.; Charatonik, Włodzimierz J. On Mazurkiewicz sets. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 817-819. http://gdmltest.u-ga.fr/item/119213/
A two-point set must be zerodimensional, Proc. Amer. Math. Soc. 116 (1992), 551-553. (1992) | MR 1093599
Planar sets that line hits twice, Houston J. Math. 23 (1997), 485-497. (1997) | MR 1690037
Sur un ensemble plan qui a avec chaque droite deux et seulement deux points communs, C.R. Soc. de Varsovie 7 (1914), 382-383. (1914)