Unit squares intersecting all secants of a square
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LetS be a square of side lengths>0. We construct, for any sufficiently larges, a set of less than 1.994s closed unit squares whose sides are parallel to those ofS such that any straight line intersectingS intersects at least one square ofS. It disproves L. Fejes Tóth's conjecture that, for integrals, there is no such configuration of less than 2s−1 unit squares.
KeywordsLine Segment Side Length Discrete Comput Geom Line intersectingS Vertical Line Segment
- [BF] I. Bárány and Z. Füredi, Covering all secants of a square,Proceedings of the Colloquia Mathematica Societatis János Bolyai, Siofok, Hungary, 1985 (in honor of L. Fejes Tóth).Google Scholar
- [MP] W. O. Moser and J. Pach, Research Problems in Discrete Geometry, Problem 84, Montréal 1985, mimeographed.Google Scholar