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Let $S\subset\mathbb R^2$ be a finite set of $m$ distinct points. List its distinct positive occurring distances as $t_1<\cdots<t_\ell$. For $1\leq j\leq\ell$, let $G_j$ be the simple graph on $S$ whose edges are the unordered pairs at distance $t_j$. Write $m_j=|E(G_j)|$ and $d_j(v)$ for the degree of $v$. Thus

$$\sum_{v\in S}d_j(v)=2m_j.$$

Indeed, an edge has two different endpoints and contributes one to each endpoint's degree. The graphs for different $j$ have disjoint edge sets. Assertions involving $t_j$ require that it occurs; the two-distance theorems assume $\ell\geq2$. We do not assign a nonexistent second distance the value zero.

Source. K. Vesztergombi, Bounds on the number of small distances in a finite planar set, Studia Scientiarum Mathematicarum Hungarica 22 (1987), 95--101, definitions on printed p. 95 (physical PDF p. 101). The whole-volume edition read is identified on the source card; the article occupies physical pp. 101--107. All result pages in this source unit use that edition.

Elementary circle facts. On a circle of radius $R$, points separated by a minor central angle $\phi\in[0,\pi]$ have chord length $2R\sin(\phi/2)$. This follows by bisecting the isosceles triangle from the center; it is strictly increasing in $\phi$ on this interval. We use degrees when writing angles such as $30^\circ$, and radians in analytic formulas involving $\pi$.

In the two-distance setting normalize $t_2=1$ and put $a=t_1\in(0,1)$. Every positive distance below $1$ between points of $S$ must equal $a$. Consequently any circular arc of angle strictly less than $60^\circ$ on a unit circle contains at most two points of $S$. Otherwise, in their order along the arc, the first point has different positive distances less than $1$ to the second and third. More generally, if points on that circle have minimum separation at least $a$, their minor angular separations are at least $\alpha=2\arcsin(a/2)$.

For points of radii $r,s$ about the same center and minor angular separation $\phi$, the squared distance is $r^2+s^2-2rs\cos\phi$, by expanding their Cartesian coordinates. These facts, finite counting and elementary trigonometric identities suffice for the local proofs here; no external theorem-level premise is imported.

Verification scope. These definitions and elementary deductions were independently checked against the published source. The final review retains that check as part of the Verified at the stated scope proof records on the [[distance_problems/vesztergombi_1987_bounds_number_small_distances_finite_planar_set/theorem_p99|second-distance theorem]], [[distance_problems/vesztergombi_1987_bounds_number_small_distances_finite_planar_set/theorem_p100|two-distance theorem]] and [[distance_problems/vesztergombi_1987_bounds_number_small_distances_finite_planar_set/construction_pp99_100|hexagonal construction]]. Multiplicity counts are not counts of distinct values or counts below an arbitrary fixed threshold.