---

Claim. The first question of Problem 132 holds for $n=7$: every set of seven points in the plane determines two distinct distances each of which occurs between at most seven pairs. Zeraoulia Rafik's note, self-published on ResearchGate under the DOI linked above and summarized in the claimant's post of 28 January 2026 in the problem's discussion thread, argues by counting. If the set determines at least four distances and all but at most one of them occur at least eight times, the $21$ pairs would number at least $1+3\cdot8=25$; a planar two-distance set has at most five points; so a counterexample determines exactly three distances, and the classification of seven-point three-distance sets, the regular heptagon or the regular hexagon with its center, is checked directly. The same counting reduces $n=8$: a counterexample would determine exactly four distances, since five or more would need at least $1+4\cdot9=37$ of the $28$ pairs and no eight-point three-distance set exists, and the Hopf–Pannwitz bound [HoPa34] on the diameter then forces the multiplicities $(1,9,9,9)$ with a unique diametral pair. The note states this reduction as a structural result and leaves the exclusion of that profile open.

Covers. The first question for $n=7$ only. The reduction of $n=8$ to the profile $(1,9,9,9)$ settles no instance by itself; the case $n=8$ is claimed by later notes that start from this reduction, on Marchetto's page and Beller's page.

Depends on. No page of this wiki.

Claimant and postings. The note was posted on 28 January 2026 in the problem's discussion thread, not on its proof-claims tab, from the account Zeraoulia Rafik; the thread post carries no AI disclosure. Marchetto's note of July 2026 names the claimant Rafik Zeraoulia, credits Zeraoulia with the counting reduction, the case $n=7$ and the $(1,9,9,9)$ reduction, and says it checked Zeraoulia's proof of $n=7$ step by step; Beller's manuscript of August 2026 and ienjoymath's note of July 2026 also credit the reduction to Zeraoulia. Those are dated uptakes by other claimants, not reviews.

Acceptance. None documented. The note is not on arXiv and has no journal record, the site labels the problem OPEN and its page does not credit the result, and no outside review is known. The claim is therefore claimed.