Bounds on the number of small distances in a finite planar set

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construction_pp99_100: Gives coordinates, an exact short-distance check and boundary counting for Vesztergombi's decorated hexagonal construction.

definitions: Fixes the occurring-distance convention and elementary circle facts used in Vesztergombi's small-distance bounds.

evidence/: Retains the scoped independent review of the local proofs and construction, its finalization delta and the preparatory reading.

lemma_1: Proves the degree bound and classifies eleven or twelve neighbors at the second smallest occurring distance.

lemma_2: Expands every high-degree and missing-vertex case proving that the endpoint degrees of a second-distance edge sum to at most twenty.

lemma_3: Proves the combined degree bound with an explicit replacement for the source's three-short-neighbor shortcut.

proposition_p96: Derives the bound m_j at most 3jm by counting graph degrees.

theorem_p100: Proves the combined multiplicity bound and its asymptotic sharpness on finite triangular-lattice patches.

theorem_p99: Uses endpoint degree sums to bound the second-distance multiplicity.

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K. Vesztergombi, Bounds on the number of small distances in a finite planar set, Studia Scientiarum Mathematicarum Hungarica 22 (1987), 95--101.

Source and version. The copy read for this card is the journal-volume scan from the Hungarian Academy repository. The article begins on printed p. 95 (physical PDF p. 101). The copy read is the whole volume; the article occupies printed pp. 95--101. The whole-volume scan prints "© Akadémiai Kiadó, Budapest" on the volume's imprint page (PDF p. 2 of 492), a notice for the volume rather than the article, every other right reserved.

Statement scope. For a finite set of $m$ planar points, let $m_j$ be the number of unordered pairs realizing its $j$th smallest distinct distance, whenever that distance occurs. The abstract and introduction on printed p. 95 state

$$m_j\leq3jm,\qquad m_2\leq5m,\qquad m_1+m_2\leq6m.$$

The introduction calls the last bound sharp for the lattice of regular triangles. The Remark on printed p. 101 gives every point of a finite piece of that lattice twelve neighbors at the first two distances, except the points on its boundary, so the sharpness is asymptotic. These are multiplicity bounds for the first few occurring distance values. They do not bound the number of distinct values, or all pairs below an arbitrary fixed numerical threshold.

Method and proof scope. The source associates a graph to each distance value and studies its vertex degrees. The earlier source check covered statements and selected method only. Complete own-words proof compilations now supply the local degree arguments, all high-degree diagram cases and the two upper bounds. The circle-counting correction in Lemma 1 and the replacement of Lemma 3's three-short-neighbor shortcut are explicitly identified as compilation-supplied. An independent source-based reviewer, distinct from the compiler, checked all the listed local proofs and repairs against the published scan named above, including the separately reviewed hexagonal construction. Their living records are Verified at the stated scope. The final review and finalization-delta review retain the reports. This is complete-proof review of those bounded scopes, not an upgrade inherited from the earlier statement check.

Compiled result scopes.

• Definitions fix unordered-pair multiplicities and the existence of the occurring values.
• Lemma 1 gives $d_j(v)\leq6j$ and classifies eleven/twelve second-distance neighbors (the paper derives the eleven-neighbor case in the proof of Lemma 2(b)); the Proposition on p. 96 gives $m_j\leq3jm$.
• Lemma 2 bounds endpoint degree sums by $20$, yielding the Theorem on p. 99, $m_2\leq5m$.
• Lemma 3 gives $d_1(v)+d_2(v)\leq12$, yielding the Theorem on p. 100, $m_1+m_2\leq6m$, with a complete triangular-patch sharpness calculation.
• The construction on pp. 99--100 has a separate proof/review scope for $m_2=(24/7)m+o(m)$, with explicit coordinates, short-distance enumeration and boundary counting.

No external theorem-level premise is used in these local proofs. Harborth's unpublished exact first-distance bound and later literature are not compiled here. Neither multiplicity theorem resolves the imported formulation of Problem 662 or its fixed-threshold variants.

Bears on. Problem 662: the Theorem on p. 100 ($m_1+m_2\leq6m$), the Theorem on p. 99 ($m_2\leq5m$) and the construction on pp. 99--100 concern the smallest-distinct-distances variant, which counts the pairs at the two smallest occurring distances of a finite set, together ($m_1+m_2$) or at the second alone ($m_2$), with no separation hypothesis and no fixed threshold $t$. The problem page records that variant as a known result, not as a reading of the problem's Statement, and Chojecki's note identifies its two-smallest-distances repair with the Theorem on p. 100. No result here bounds the number of pairs at distance at most a fixed $t$, so none bears on the Statement as worded.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.