Isosceles planar subsets (Q1387849)

A planar set is \(k\)-isosceles if every \(k\)-point subset contains a 3-set in which one point is equidistant from the other two. The author and Paul Erdős showed recently that the unique largest 3-isosceles set has 6 points, namely the center and vertices of a pentagon. Here, the author characterizes the 3-isosceles 5-sets, and makes significant progress on the classification of 4-isosceles sets. Eleven 4-isosceles 8-sets are exhibited, and it is conjectured that there are no 9-sets. It is shown that a 4-isosceles 8-set cannot have five points on a line, and that there are exactly two 4-isosceles 7-sets with five points on a line, and a unique 4-isosceles 8-set with 4 points on a line. It is shown that a 4-isosceles set that contains the vertices of a square can have at most 7 points; thus no such set is maximal. The paper concludes with a half-dozen elementary, interesting, and probably difficult conjectures.