Least cardinality of a set of points in the plane - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:29:30Z http://mathoverflow.net/feeds/question/90454 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90454/least-cardinality-of-a-set-of-points-in-the-plane Least cardinality of a set of points in the plane Holowitz 2012-03-07T13:46:59Z 2012-03-07T17:34:02Z <p>What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all the euclidean distances from X to points of S are rational?</p> <p>Can we do better then $3\leq K \leq 2^{\aleph_0}$?</p> <p>What can be said about K if all the points of S must be at integer distance from eachother, can it be proved to be finite?</p> http://mathoverflow.net/questions/90454/least-cardinality-of-a-set-of-points-in-the-plane/90479#90479 Answer by Tony Huynh for Least cardinality of a set of points in the plane Tony Huynh 2012-03-07T16:41:18Z 2012-03-07T17:34:02Z <p>As Boris Bukh points out, three points suffice, but I'd like to point out that your question is related to this <a href="http://mathoverflow.net/questions/54104/drawing-planar-graphs-with-integer-edge-lengths" rel="nofollow">MO question</a>. </p> <p>Here is a summary of the information in the previous question. For the second part of your question, the author (me) conjectures that for any finite set $S$ with all rational distances, no such point $P$ exists. As I noted in the comments, this is true when $|S|=3$, proven by Almering.</p> <p>It is not known if there is a point with all rational distances to the unit square. However, it is known that there are no points at rational distance from all vertices of a regular $n$-gon, except perhaps when $n=4,6,8,12,24$. </p> <p>Some more tangential remarks are that it is not known if there is a dense set of points in the plane with all distances rational, although it is conjectured that there is none.</p> <p>Even more tangential, it is not known if every planar graph can be straight-line embedded in the plane with all edges having rational length, although it is conjectured to be true. </p>