#132: the second low-multiplicity distance in convex position

The target (erdosproblems.com/132, OPEN). For a distance d determined by a planar n-point set A, let μ_A(d) count the unordered pairs at distance d. Erdős asked whether, for every n≥5, there must be two distinct distances with multiplicity at most n, and whether the number of such distances tends to infinity with n. The Hopf-Pannwitz theorem supplies one: the diameter occurs at most n times. The problem is to find another.

The convex-position theorem. Clemen, Dumitrescu, and Liu proved the first question when every point of A is a vertex of its convex hull:

If A is a convex n-point set with n≥5, then A determines a distance d with
0<d<diam(A) and 1≤μ_A(d)≤n.

The count. Every convex n-point set determines at least ⌊n/2⌋ distinct distances. If it determines more and every non-diameter distance occurs at least n+1 times, then at least ⌊n/2⌋ non-diameter classes plus one diameter pair would force

C(n,2)≥⌊n/2⌋(n+1)+1>C(n,2),

a contradiction. The excess in the middle expression is exactly n+1 for even n and (n+1)/2 for odd n.

The equality cases. The remaining case is the Altman-Fishburn classification at exactly ⌊n/2⌋ distances. A regular odd n-gon has every chord-step class n times. A regular even n-gon has multiplicity vector (n,...,n,n/2). Deleting one vertex from a regular odd (n+1)-gon leaves every class n-1 times. At n=6 the possible vectors are (6,6,3) and (5,5,5). Each model has at least two distance classes and every class has multiplicity at most n, so one non-diameter class works.

Source and audit. This is Theorem 1.2 of the published 2025 paper. The independent audit checked the Version of Record and arXiv source, recovered Fishburn's full 29-page classification paper, and verified the page and line locators. Exact-integer code checks 9,996 pair-count cases, 496 regular models, 247 deleted-vertex models, and both hexagon vectors under normal Python, -O, and -OO, with controlled negative tests. The public package records source URLs, hashes, and licenses but redistributes no third-party PDF or site snapshot.

Scope. This is a literature-derived partial, not a solution of Problem 132. It does not cover arbitrary planar point sets and does not address the stronger growth question. On the audit date the live problem was OPEN with five informal comments, zero registered or dedicated proof claims, and no current worker; none of those comments enters this proof.