On Erd(?)s' Problems About Finite Planar Sets

The study about geometric properties of finite sets of points in the plane is one of the most famous and fundamental topics in combinatorial geometry. Many Erdos' problems are about this topic. The aim of this work is to discuss some Erdos' problems of finite planar sets.A finite planar set is k-isosceles for k ≥ 3, if every k-point subset of the set contains a point equidistant from other two. P. Fishburn specified 3-isosceles planar sets, obtained several important results about 4-isosceles planar sets and posed a series of conjectures and open questions (see [47]). In Chapter 1, we disprove the Conjecture 1 in [47] and give answers to 4 open questions about 4-isosceles sets: there exists a 4-isosceles 6-point set F with no 4 points on a circle and no 3 on a line: there exists a 4-isosceles 7-point set F with no 4 points on a circle; there exist at least two 4-isosceles 9-point sets, besides the sets shown in Fig. 1 in [47], thirty four 4-isosceles sets with 8 points are noted; and a 4-isosceles 8-point set which has four points on a line is given.A set of n points in the plane is said to be in convex position, if none of its elements is contained in the interior of a triangle induced by three others. A set of n points in the plane is said to be in strictly convex position, if none of its elements is contained in the convex hull of the others. Janos Pach and Rom Pinchasi (see [61]) proved that any set of npoints in strictly convex position in the plane has at most [2(n - 1)/3J triples that induce unit triangles (equilateral triangles of side length one), and any set of n points in convex position in the plane has at most n - 2 triples that induce unit triangles. [61] also gives some results about general triangles. In Chapter 2, we focus on isosceles right triangles. At most 2n-4 congruent isosceles right triangles can be induced by a set of n points in convex position, in strictly convex position at most n congruent isosceles right triangles can be induced, and the two bounds are tight. Furthermore, any set of n points in convex position in the plane induces at most 2n - 2 congruent copies of a fixed isosceles triangle.In [6] Andras Bezdek proved that if a convex n-gon and n points are given, then the points and the sides of the polygon can be renumbered so that at least [n/3] triangles spanned by the ith point and the ith side (i = 1, 2,… n) are mutually non-overlapping. In Chapter 3. we show that at least [n/2] mutually non-overlapping triangles can be constructed. This lower bound is best possible.