Independent record of Erdős Problem 132

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Status and scope

The targets are (i) $R(A)\ge2$ for every set of $n\ge5$ distinct planar points, and (ii) $\min_{|A|=n}R(A)\to\infty$. The threshold in both questions is the size of the full set, not the size of a selected subset.

Neither assertion for all finite planar sets is resolved in this investigation. No planar counterexample is constructed. The results below are proved for the indicated hypotheses; no passage from a restricted family to arbitrary configurations is assumed.

All multiplicities count unordered pairs, and

$$m_A(r)=\#\{\{p,q\}\subset A:|p-q|=r\},\qquad R(A)=\#\{r:1\le m_A(r)\le |A|\}.$$

Only the supplied problem page and source files were used. No browsing was performed. The mention of unavailable earlier notes in the problem page was not used as mathematical evidence.

The principal conclusions are:

1. Arbitrary sets, finite orders: $R(A)\ge2$ for every $5\le |A|\le10$, by exact counting and the supplied few-distance classifications.
2. Arbitrary sets, a quantitative layer bound: if $h$ is the largest number of vertices in any convex layer, then

$$\boxed{R(A)\ge\max\left\{1,\left\lfloor\sqrt{\frac{n}{2h}}\right\rfloor\right\}.}\tag{1}$$

More precisely, writing $s_k$ for the number of points in the first $k$ layers, $2ks_k\le n$ implies that the $k$ largest distances are rare. 3. A restricted asymptotic theorem: $R(A_n)\to\infty$ if every fixed number of outer convex layers contains $o(|A_n|)$ points. In particular this holds if the largest layer is $o(n)$, or if diameter divided by minimum separation is $o(n)$. 4. An independently proved affine restriction: if a nonsimilarity invertible affine map preserves all equal-distance relations of $A$, then

$$R(A)\ge\left\lceil\frac{n-1}{4}\right\rceil,$$

and $R(A)\ge2$ throughout the required range $n\ge5$.

This record contains the principal proofs, exact coordinate calculations, failed approaches, and explicit uncertainties. Sections 1–5 give the main results and the unresolved general gap; Appendix A gives finite configurations and remaining small-order cases; Appendix B records the affine obstruction; Appendix C gives the abstract countermodel’s Euclidean obstruction and the component argument; Appendix D records superseded and unsuccessful approaches; Appendix E states the final limits. The investigation's scratch files are not retained; this record is intended to be readable without them, and the supplied published classifications remain explicit external inputs.

1. Largest-distance ranks and convex depth

Delete the vertices of the convex hull repeatedly to obtain layers $L_1,L_2,\ldots$. For a collinear remaining set, its hull vertices are its two endpoints, or its sole point. Write $\ell(p)=i$ when $p\in L_i$, and

$$H_k=L_1\cup\cdots\cup L_k,\quad s_k=|H_k|, \qquad d_1>d_2>\cdots>d_q>0$$

for the distinct distances in decreasing order.

Lemma 1: depth restricts distance rank

If $|p-q|=d_j$, then

$$\ell(p)+\ell(q)\le j+1.\tag{2}$$

Proof. Suppose $p$ has depth $i>1$. It is a convex combination of at least two distinct vertices of $L_{i-1}$. Strict convexity of $x\mapsto|x-q|^2$ implies that at least one of those vertices $p'$ satisfies $|p'-q|>|p-q|$. Repeat until the first endpoint is on $L_1$, obtaining $i-1$ strict increases. Keep the resulting endpoint fixed and move the other endpoint outward in the same way, obtaining $\ell(q)-1$ further strict increases. These yield $i+\ell(q)-2$ distinct distances larger than $|p-q|$, so there can be at most $j-1$ of them. A replacement never coincides with the fixed endpoint, since the new distance strictly increases. This proves (2), including degenerate layers. $\square$

In particular, all pairs at one of the first $j$ distances have both endpoints in $H_j$. Thus

$$m_A(d_j)=m_{H_j}(d_j),\tag{3}$$

and $d_j$ retains its rank in $H_j$.

Lemma 2: a rank-dependent linear bound

If a distance $r$ has at most $k-1$ distinct larger distances in a planar set $B$, its distance graph is $2k$-degenerate. In particular,

$$m_B(r)\le2k|B|.\tag{4}$$

Proof. Consider any nonempty subset $T\subseteq B$, and translate a leftmost point $p$ to the origin. All points of $T$ lie in the closed right halfplane. If $p$ has at most $2k$ neighbors at distance $r$, it is the required low-degree vertex.

Otherwise its $t\ge2k+1$ neighbors can be written

$$q_j=r(\cos\theta_j,\sin\theta_j),\qquad -\pi/2\le\theta_1<\cdots<\theta_t\le\pi/2.$$

Choose $q_i$ with at least $k$ of these neighbors on each angular side. We claim that $q_i$ has no distance-$r$ neighbor in $T$ other than $p$.

Suppose such a neighbor $x\ne p$ exists. Write

$$x=\rho(\cos\phi,\sin\phi),\quad \rho>0, \quad -\pi/2\le\phi\le\pi/2.$$

The equality $|x-q_i|=r$ gives

$$\cos(\theta_i-\phi)=\frac{\rho}{2r}>0.$$

If $\theta_i\ge\phi$, use $k$ indices with $\theta_j>\theta_i$. The angles $\theta_j-\phi$ are distinct and lie in $(\theta_i-\phi,\pi]$. Since cosine is strictly decreasing on $[0,\pi]$, the quantities

$$|x-q_j|^2=\rho^2+r^2-2\rho r\cos(\theta_j-\phi)$$

are distinct and strictly greater than $r^2$. This gives $k$ distinct larger distances, a contradiction. If $\theta_i\le\phi$, use the $k$ neighbors below $\theta_i$ instead, with angles $\phi-\theta_j$. Hence $q_i$ has degree one.

Every induced subgraph therefore contains a vertex of degree at most $2k$. Removing such vertices successively counts every unordered edge once and proves (4). Ties in the leftmost coordinate cause no problem: the angles still lie in a closed semicircle. $\square$

Theorem 3: the layer bound

If $A$ has at least $k$ distances and $2ks_k\le n$, then $R(A)\ge k$; specifically, its $k$ largest distances are rare.

Proof. For every $j\le k$, equations (3) and (4) give

$$m_A(d_j)=m_{H_j}(d_j)\le2js_j\le2ks_k\le n.$$

These classes are nonempty by definition. $\square$

Let $h=\max_i|L_i|$. Since $s_k\le kh$, this proves (1), apart from checking that the required distances exist. If there are $t$ layers, then $t\ge n/h$. Applying Lemma 1 to a deepest point and any other point shows that there are at least $t$ distances. Consequently $\lfloor\sqrt{n/(2h)}\rfloor\le t\le q$. The additional lower bound one is the diameter theorem: the diameter occurs on at most $n$ unordered pairs. Its statement and proof were read in Erdős (1946), Section 4.

For a sequence with $s_k=o(n)$ for each fixed $k$, Theorem 3 proves $R(A_n)\to\infty$. There are eventually at least $k$ distances: if there are only $q$, two fixed distinct centers and intersections of their distance circles give $n\le2q^2+2$.

A geometric consequence in terms of separation

Let $\delta$ be the least distance and $\Delta$ the diameter. A layer with at least three vertices has perimeter at least $|L_i|\delta$ and at most $\pi\Delta$, so

$$h\le\frac{\pi\Delta}{\delta}.$$

The same inequality holds for one- or two-point layers. For completeness, the perimeter upper bound follows by integrating projections: for a convex polygon, the sum of absolute projections of its edges on a line is twice its width in that direction. Integrating over directions $0\le\theta\le\pi$, each edge of length $a$ contributes $2a$, while every width is at most $\Delta$. Hence its perimeter is at most $\pi\Delta$.

Thus

$$\boxed{R(A)\ge\max\left\{1, \left\lfloor\sqrt{\frac{n\delta}{2\pi\Delta}}\right\rfloor\right\}.}\tag{5}$$

In particular, uniformly separated sets of diameter $O(\sqrt n)$ have $R(A)=\Omega(n^{1/4})$, by this elementary argument.

There is also a complementary elementary estimate. Two vertices of a fixed-distance graph have at most two common neighbors. If that graph has $s$ vertices and $e$ edges, counting pairs of neighbors and using Cauchy–Schwarz gives

$$\sum_v\binom{\deg v}{2}\le2\binom{s}{2}, \qquad e\le U(s):=\frac{s}{4}\bigl(1+\sqrt{8s-7}\bigr).$$

Hence $U(s_k)\le n$ is another sufficient condition for the first $k$ distances to be rare. For instance $s_k\le n^{2/3}$ suffices, since $U(s)\le s/4+s^{3/2}/\sqrt2$ and $1/4+1/\sqrt2<1$. This also gives $R(A)\ge\lfloor n^{2/3}/h\rfloor$. If one invokes the general unit-distance bound $m_B(r)\le C|B|^{4/3}$ stated in the supplied Clemen–Dumitrescu–Liu Section 1.4, the same support argument improves this supplementary bound to $R(A)\ge\lfloor c n^{3/4}/h\rfloor$ for an absolute $c>0$. The rank-dependent bound (1) and its consequences do not need that incidence theorem.

2. Exact consequences for small orders

Let $q$ be the number of distinct distances and $N=\binom n2$. If $R(A)=1$, its sole rare distance is the diameter, and

$$N\ge1+(q-1)(n+1),\qquad q\le\lfloor n/2\rfloor.\tag{6}$$

If $n=2t$ and $q=t$, equality holds: the diameter occurs once and every other distance occurs exactly $n+1$ times. Deleting either endpoint of the diameter leaves exactly $t-1$ distances, each with multiplicity at most $n+1$.

Let $g(k)$ denote the maximum cardinality of a planar set determining exactly $k$ distances. Theorem 1 of the supplied Erdős–Fishburn (1996) paper gives

$$g(2)=5,\quad g(3)=7,\quad g(4)=9,$$

and classifies the sets attaining these bounds. Its statement and relevant proof passages were read. The classifications, with exact multiplicities, are:

The seven-point types are a regular heptagon and a regular hexagon with its center. For nine points the source lists exactly four similarity types. Because its figures are only textually reproduced, Appendix A.1 below constructs four pairwise nonsimilar coordinate models and computes all their distances exactly. This exhausts the classification without guessing missing figure coordinates.

For $n=5,7,9$, (6) and the bounds on $g$ force respectively two, three, and four distances. The table excludes $R=1$.

For $n=6,8,10$, (6) forces respectively three, four, and five distances, so the even-order equality case applies. Delete either endpoint of the unique diameter. For $n=6$, both resulting sets are regular pentagons. For $n=8$, both are regular heptagons, since the alternative seven-point type has multiplicity twelve, greater than $n+1=9$. For $n=10$, both are regular nonagons, since every nonregular nine-point type has a multiplicity at least fifteen, greater than eleven.

In each case two regular $(n-1)$-gons share $n-2\ge4$ vertices. Three shared vertices determine their circumcircle, and one vertex on that circle determines the regular polygon of a specified order. Thus the two sets coincide, contradicting deletion of distinct endpoints. This proves $R(A)\ge2$ for every $5\le n\le10$.

The exact calculations in Appendix A.2 establish:

• Every twelve-point five-distance set has multiplicities $(24,15,12,12,3)$ in increasing distance order, using Shinohara (2008), Theorem 1.2(b), and an explicit lattice model. Thus it has three rare distances.
• Deleting one specified point of that model gives eleven points with multiplicities $(18,12,10,12,3)$, and exactly two rare distances.
• A possible counterexample at $n=11$ must be a five-distance set. A possible counterexample at $n=12$ must have six distances with multiplicities $(13,13,13,13,13,1)$, the last class being the diameter. Neither possibility is excluded by the classifications used here.

3. A linear bound under a precise affine hypothesis

Suppose an orthonormal coordinate system has the property

$$|p-q|=|p'-q'|\quad\Longrightarrow\quad |x(p)-x(q)|=|x(p')-x(q')|.\tag{7}$$

Then equality of absolute vertical displacements follows as well. Thus each distance has a fixed displacement pair $(a,b)$, up to signs.

Every vertical distance class is a disjoint union of finite paths on the vertical fibers, so it has at most $n-1$ edges. Write $m=\min x$, $M=\max x$, and $W=M-m>0$; the case $W=0$ is collinear and immediate. A class with $a>W/2$ has maximum degree at most two: a point cannot have neighbors on both lines $x-a$ and $x+a$, and the one possible neighbor line allows at most two ordinates. Such a class is therefore rare.

Let $c$ points lie on $x=(M+m)/2$. Of the remaining points, at least $(n-c)/2$ lie strictly on one side of that line. Join these to an anchor on the opposite extreme line. Every resulting distance is rare by the preceding paragraph, and at most two anchor edges have the same distance by (7). We obtain at least $\lceil(n-c)/4\rceil$ rare classes. If $c\ge1$, the middle fiber gives another $c-1$ distinct vertical distances; by (7), these are globally rare and distinct from the previous classes. Therefore

$$R(A)\ge\left\lceil\frac{n-1}{4}\right\rceil.\tag{8}$$

For $c=0$, the anchor argument gives the stronger $\lceil n/4\rceil$.

For $n\ge6$, (8) gives at least two. At $n=5$, every class is rare: a non-axis class has edge vectors $\pm(a,b),\pm(a,-b)$, so each connected component is a square-grid subgraph. If a component has $u$ vertices in $r$ rows and $s$ columns, it has at most

$$2u-r-s\le2u-2\sqrt u$$

edges. Summing gives at most $2n-2\sqrt n$, whose floor is five at $n=5$. Axis classes are unions of paths. Five planar points cannot all be mutually equidistant, so at least two classes occur.

Condition (7), in some orthonormal coordinates, is equivalent to the existence of an invertible nonsimilarity affine map preserving every original equal-distance relation. Indeed, write its positive definite matrix $T^{\mathsf T}T$ in an orthonormal eigenbasis with distinct eigenvalues $\lambda,\mu$. For equal-length vectors $d,e$, the two equalities

$$d_x^2+d_y^2=e_x^2+e_y^2, \qquad \lambda d_x^2+\mu d_y^2=\lambda e_x^2+\mu e_y^2$$

force $d_x^2=e_x^2$. Conversely, (7) permits any positive diagonal stretching in those coordinates.

This hypothesis is a real restriction. An equilateral triangle, or a square, forces every equality-preserving affine map to be a similarity; the matrix obstruction is detailed in Appendix B. A generic perturbation cannot be substituted, because it splits old distance classes and can turn a rich original class into many rare new ones.

4. A concrete obstruction to diameter-only arguments

Even imposing the diameter edge bound, absence of even cycles, and at most one cycle on every induced maximum-color graph does not force two rare colors in an ordered edge-coloring of a complete graph.

For $n=4s$, $s\ge2$, label vertices $0,\ldots,n-1$ and color a pair according to which of these intervals contains $|a-b|$:

$$\{1\},\quad[2,3],\quad[4,5],\quad\ldots,\quad [2s-2,2s-1],\quad[2s,4s-1].$$

Order the colors as displayed, with $\{1\}$ highest. Its multiplicity is $n-1$. A middle color $[2j,2j+1]$ has

$$(n-2j)+(n-2j-1)=2n-4j-1\ge n+3$$

edges, and the last color has $s(2s+1)>n$. There is exactly one rare color and $n/4+1$ colors in total.

In any subset whose highest occurring color is $[L,U]$, all consecutive label gaps are at least $L$. Nonconsecutive labels differ by at least $2L$, whereas every displayed interval satisfies $U<2L$. Thus the highest-color graph is a subgraph of the path through consecutive labels. Every induced maximum-color graph is a linear forest, yet only one global color is rare.

These are not Euclidean counterexamples. Their restriction to five consecutive vertices is already unrealizable, as proved by the explicit circle-intersection equations in Appendix C.1. The construction shows precisely that inter-color Euclidean constraints, beyond hereditary diameter edge counts or abstract diameter graph structure, are indispensable.

5. Remaining gap

The layer theorem gives a precise necessary condition for a sequence with bounded $R$. If $R(A)\le M$, $n>2M^2+2$, and $K=M+1$, then $A$ has at least $K$ distances. Theorem 3 forces

$$s_K>\frac{n}{2K}.$$

At least one of its first $K$ convex layers consequently has more than $n/(2K^2)$ vertices. Thus any bounded-$R$ sequence must have a macroscopic convex layer at bounded depth.

The supplied Clemen–Dumitrescu–Liu paper proves $R\ge2$ for convex sets and proves

$$m_A(d_2)\le\min\left\{ \tfrac32(|L_1|+|L_2|),\ \tfrac43|L_1|+2|L_2|,\ 2|L_1|+|L_2|\right\}.$$

Its statements and proofs in Sections 1.1 and 2 were read. These yield additional sufficient conditions, but do not fill the macroscopic-layer gap. In the displayed derivation of the last bound, the supplied transcription has an arithmetic typo: after $e(L'_1,L'_2)=2|L'_2|$, the bound on $e(L'_1)$ is $2|L'_1|-|L'_2|$, not $2|L'_1|-2|L'_2|$. Its following display uses the correct coefficient.

Nor does the convex theorem alone establish divergence for arbitrary convex sets. In any configuration with $q$ distances,

$$\binom n2\ge(q-R)(n+1)+R, \quad\text{so}\quad q\le\frac{\binom n2+nR}{n+1}.$$

For convex sets, the supplied distinct-distance lower bound is $q\ge\lfloor n/2\rfloor$. Hence bounded $R$ forces $q=n/2+O(1)$, but no classification or stability theorem for this entire bounded-excess regime is established here.

The unresolved tasks are therefore substantive: control rare classes in configurations with large outer layers, or find another inter-color Euclidean mechanism. Passing to a convex subset does not preserve the required multiplicity upper bound after the other points are restored. Passing to a generic deformation does not preserve rare classes when equalities split. Neither transfer is used in the proved results above.

Source passages used

• Erdős (1946), On sets of distances of n points, Section 4: the diameter bound.
• Erdős–Fishburn (1996), Maximum planar sets that determine k distances, Theorem 1 and Sections 2–3: extremal sizes and the classifications used for $5\le n\le10$.
• Shinohara (2008), Uniqueness of maximum planar five-distance sets, Theorems 1.1–1.2 and Section 3: uniqueness at twelve points and the stated diameter-graph restrictions.
• Clemen–Dumitrescu–Liu (2025), On multiplicities of interpoint distances, Sections 1.1 and 2: the convex and two-layer results. Their Section 1.4 was also read, but its reported unit-distance incidence bound is not needed for the elementary main results here.
• Vesztergombi (1987), On large distances in planar sets: the second-largest-distance bound and its proof were read while examining extensions.
• Bhowmick, A problem of Erdős about rich distances, Section 2: the common-vertex rotation and common-edge reflection constructions, recorded as an unsuccessful route to controlling the number of rare classes.

All these refer to the cited papers. No assertion above depends on an unavailable figure, an external web source, or an assumed earlier investigation.

Appendix A. Exact configurations and unresolved finite cases

A.1. Four nine-point four-distance models

Let $P_m$ denote a regular $m$-gon. Its $(m-1)/2$ distances each occur $m$ times when $m$ is odd. Thus $P_9$ has the multiplicity vector

$$(9,9,9,9).$$

For the next two models use axial triangular-lattice coordinates. A pair $(x,y)\in\mathbb Z^2$ represents the Euclidean point

$$\phi(x,y)=\left(x+\frac y2,\frac{\sqrt3\,y}{2}\right).$$

For coordinate difference $(u,v)$, the squared Euclidean distance is

$$Q(u,v)=u^2+uv+v^2. \tag{A1}$$

Put

$$H=\{(0,0),(1,0),(0,1),(-1,1),(-1,0),(0,-1),(1,-1)\},$$

and define the coordinate sets

$$\begin{aligned} T_1={}&H\cup\{(1,1),(-1,2)\},\\ T_2={}&\{(0,0),(1,0),(0,1),(-1,1),(-1,2),\\ &\hspace{31mm}(1,1),(2,0),(1,-1),(0,-1)\}. \end{aligned}$$

Both $\phi(T_i)$ have squared-distance set $\{1,3,4,7\}$. Their multiplicities in this order are

$$\begin{array}{c|rrrr} &1&3&4&7\\ \hline \phi(T_1)&16&9&7&4\\ \phi(T_2)&15&9&6&6 \end{array} \tag{A2}$$

For an explicit integer check of (A2), the distance-degree vectors of vertices, in squared-distance order $(1,3,4,7)$, have the following distributions. Summing each coordinate and dividing by two gives the unordered pair counts.

For the final model identify the plane with $\mathbb C$, put $s=\sqrt3$, and let $\omega=e^{2\pi i/3}$. Set

$$C=\{a\omega^j:a\in\{1,1+s,-(2+s)\},\ j=0,1,2\}.$$

The three signed radii have distinct absolute values, so these are nine distinct points. For real $a,b$, pairs with the same exponent have squared distance $(a-b)^2$, while pairs with distinct exponents have squared distance $a^2+ab+b^2$. There are three pairs of the first kind and six of the second between two distinct three-point orbits. Within the orbit of radius $a$, there are three pairs of squared distance $3a^2$.

Writing $a=1$, $b=1+s$, $c=-(2+s)$, the exact table is

$$\begin{array}{c|c|c} \text{pairs}&\text{squared distance}&\text{number}\\ \hline \text{within }a&3&3\\ \text{within }b&12+6s&3\\ \text{within }c&21+12s&3\\ a,b\text{; same exponent}&3&3\\ a,b\text{; distinct exponents}&6+3s&6\\ a,c\text{; same exponent}&12+6s&3\\ a,c\text{; distinct exponents}&6+3s&6\\ b,c\text{; same exponent}&21+12s&3\\ b,c\text{; distinct exponents}&6+3s&6 \end{array}$$

Hence $C$ has four squared distances, in increasing order,

$$3,\quad6+3\sqrt3,\quad12+6\sqrt3,\quad21+12\sqrt3,$$

with corresponding multiplicities

$$(6,18,6,6). \tag{A3}$$

The four models have distinct sorted multiplicity multisets:

$$(9,9,9,9),\ (16,9,7,4),\ (15,9,6,6),\ (18,6,6,6).$$

They are therefore pairwise nonsimilar. The nine-point classification implies that they exhaust all nine-point four-distance sets. In particular, every such set has at least three distance classes of multiplicity at most nine. Also, each nonregular type contains a distance of multiplicity at least fifteen.

A.2. The twelve-point model and an eleven-point example

In the same axial coordinates, define

$$\begin{aligned} U=\{&(0,1),(0,2),(0,3),\\ &(1,0),(1,1),(1,2),(1,3),\\ &(2,0),(2,1),(2,2),(3,0),(3,1)\}. \end{aligned}$$

Using (A1), its squared-distance set and multiplicities are

$$\begin{array}{c|rrrrr} \text{squared distance}&1&3&4&7&9\\ \hline \text{multiplicity in }\phi(U)&24&15&12&12&3. \end{array} \tag{A4}$$

Here is a grouped exact check. The distance-degree vectors in the order $(1,3,4,7,9)$ are:

Their sum is $(48,30,24,24,6)$, proving (A4). Shinohara's uniqueness theorem implies that every twelve-point five-distance set is similar to this one. Thus each has exactly three rare distances at threshold twelve.

Deleting $(1,1)$ from $U$ gives an eleven-point set with multiplicity vector

$$(24,15,12,12,3)-(6,3,2,0,0) =(18,12,10,12,3).$$

It has exactly two rare distances at threshold eleven.

Every eleven-point subset of $\phi(U)$ has at least two distances occurring at most eleven times: squared distance four has multiplicity ten because all vertices have degree two in that class; squared distance nine has multiplicity two or three. Also, every such subset retains at least $24-6=18$ pairs at the shortest distance.

A.3. Remaining cases at eleven, twelve, and thirteen points

The use of thirteen points below also invokes the source value $g(5)=12$, from Erdős–Fishburn (1996), Theorem 1.

For $n=11$, a hypothetical counterexample must have exactly five distances. The four nondiameter distances have multiplicity at least twelve and the diameter has multiplicity at most seven. The diameter graph cannot have a one-vertex cover: deleting that vertex would leave ten points with at most four distances, contradicting $g(4)=9$. The unique twelve-point classification does not classify all eleven-point five-distance sets, so Appendix A.2 does not exclude this case.

For $n=12$, a hypothetical counterexample has either five or six distances by the counting bound (6) and $g(4)=9$. The five-distance case is excluded by (A4). In the six-distance case the even-order equality argument forces five multiplicities thirteen and a unique diameter. Deleting either endpoint gives an eleven-point five-distance set with every multiplicity at most thirteen. Such a set cannot be contained in a twelve-point five-distance set, because every eleven-point subset of the unique twelve-point model retains a multiplicity at least eighteen. If both endpoint deletions were regular hendecagons, their ten shared vertices would force them to coincide. Therefore at least one is a nonregular eleven-point five-distance set with maximum multiplicity at most thirteen, not extendible to any twelve-point five-distance set. The supplied classification results do not exclude such an eleven-point set.

For $n=13$, a hypothetical counterexample has exactly six distances, with five nondiameter multiplicities at least fourteen and diameter multiplicity at most eight. Its diameter graph cannot have a one-vertex cover. Otherwise deleting that vertex leaves a twelve-point five-distance set and hence a class of multiplicity at least twenty-four. Then the original thirteen-point set has at least

$$24+4\cdot14+1=81>\binom{13}{2}=78$$

pairs, a contradiction. This is only a necessary condition, not a resolution for thirteen points.

The uncontrolled cases involve arbitrary eleven-point five-distance sets or larger few-distance sets, and no classification or extension principle for them has been proved here.

Appendix B. Affine obstruction and failed deformation transfer

B.1. A finite linear-algebra test

There is also a finite-dimensional test. For every pair of equal-length difference vectors $d,e$, form the traceless symmetric matrix $dd^{\mathsf T}-ee^{\mathsf T}$. Let $V$ be their span in the two-dimensional space of traceless symmetric $2\times2$ matrices. The equivalent conditions hold exactly when $\dim V\le1$. Indeed, a nonzero traceless symmetric $S$ orthogonal to $V$ gives a positive definite nonscalar $Q=tI+S$ for sufficiently large $t$; conversely, the traceless part of any admissible nonscalar $Q$ is a nonzero vector orthogonal to $V$.

B.2. Explicit obstruction and safe direction of transfer

An equilateral triangle prevents the nonsimilarity affine condition of Section 3. In coordinates with unit edge vectors $(1,0),(1/2,\sqrt3/2),(-1/2,\sqrt3/2)$, preserving equality of their transformed squared lengths forces the off-diagonal entry of $Q$ to vanish and its two diagonal entries to agree. A square also prevents the nonsimilarity affine condition of Section 3: equality of horizontal and vertical side lengths forces $q_{11}=q_{22}$, and equality of its two diagonal lengths forces $q_{12}=0$. Thus even these elementary configurations force $Q$ to be scalar.

Arbitrary perturbation does not repair this obstruction. It usually splits an original distance class into smaller classes; these new classes can be rare even when the original class was rich. Hence a lower bound for $R$ after a generic perturbation does not transfer back. For example, six vertices of a regular unit hexagon together with its center give seven points with multiplicities $12,6,3$, and thus $R=2$. A generic sufficiently small perturbation gives 21 distinct distances, all rare.

In contrast, if an injective deformation preserves every original equal-distance relation, each new class is a union of original classes. Every original class inside a rare new class is itself rare, so $R(\text{original})\ge R(\text{deformed})$. This is the safe direction of transfer, but existence of such a useful deformation for arbitrary sets remains unproved. The affine examples above show that it cannot always be achieved by a nonsimilarity affine map.

Appendix C. Failed abstract and component arguments

C.1. Five-point nonrealizability of the ordered coloring

These are not planar counterexamples. Their restriction to the five vertices $0,1,2,3,4$ already cannot be realized by planar points respecting the color order.

Suppose it could be realized by $p_0,\ldots,p_4$, with distances $a>b>c>0$ for colors $0,1,2$. Then adjacent labels are at distance $a$, labels whose difference is two or three are at distance $b$, and $|p_0p_4|=c$.

Put $p_1=(0,0)$ and $p_2=(a,0)$. The conditions on $p_0$ and $p_3$ give

$$p_0=\left(a-\frac{b^2}{2a},y\right), \qquad p_3=\left(\frac{b^2}{2a},z\right), \qquad y^2=z^2=b^2-\frac{b^4}{4a^2}>0.$$

If $z=-y$, then

$$|p_0p_3|^2= \left(a-\frac{b^2}{a}\right)^2+4y^2 =a^2+2b^2>b^2,$$

contrary to $|p_0p_3|=b$. Hence $z=y$, and $p_0,p_3$ are reflections of each other across the perpendicular bisector of $p_1p_2$. Since $|p_1p_4|=|p_2p_4|=b$, the point $p_4$ lies on this perpendicular bisector. Therefore

$$|p_0p_4|=|p_3p_4|=a,$$

contradicting $|p_0p_4|=c<a$.

The obstruction is thus an exact relation between different distance colors, not the numerical or abstract graph structure of diameter classes.

C.2. A component criterion without a counting conclusion

For an actual planar point set $A$ and a distance $r$, form the graph $G_r$ with vertex set $A$ and edges at distance $r$. If every nontrivial connected component of $G_r$ has geometric diameter at most $r$, then $m_A(r)\leq|A|$: within each such component, every $r$-edge is a diameter edge, so the planar diameter bound applies separately; summing the vertex counts gives the claim.

Equivalently, every rich distance $r$ has a connected component containing two points at some distance greater than $r$. One can strengthen the statement slightly by taking a component with positive graph excess $e-v$: its geometric diameter must be greater than $r$.

This yields an upward dependency from each rich distance to a larger distance. It does not give an injection into rare distances, a branching bound, or any useful restriction on how many rich distances can depend on the same larger distance. No such missing assertion is established here.

Appendix D. Superseded and unsuccessful approaches

This section records work already undertaken. It does not propose additional assertions as established results.

D.1. A valid but superseded sampling bound for the largest distances

Before the direct angular proof, an independent-set sampling argument gave an exponential-in-rank bound. It is retained because it isolates what can be obtained using only hereditary diameter counting.

Let $F_k(B)$ be the graph consisting of pairs at the $k$ largest distances of $B$, or all its pairs if it has fewer than $k$ distances. One can prove

$$|E(F_k(B))|\le C_k|B|,\qquad C_1=1,\qquad C_k=(1+4\mathrm e)C_{k-1}+\mathrm e,$$

where $\mathrm e$ denotes Euler's number. Equivalently,

$$C_k=\frac{5(1+4\mathrm e)^{k-1}-1}{4}.$$

For the inductive step, let $r$ be the $k$-th largest distance and let $G$ consist of all pairs longer than $r$. The inductive bound holds on every induced subset, since its edges lie among at most the $k-1$ largest distances of that subset. Hence every induced subgraph of $G$ has a vertex of degree at most $2C_{k-1}$. Orient its edges according to a successive low-degree deletion order; every outdegree is at most $2C_{k-1}$.

Independently mark vertices with probability $p$, and retain a marked vertex only if all its outneighbors are unmarked. The retained set $S$ is independent in $G$, so all its distances are at most $r$, and its distance-$r$ graph has at most $|S|$ edges by the diameter theorem. For an original distance-$r$ edge, its endpoints are not adjacent in $G$, so both are retained with probability at least

$$p^2(1-p)^{4C_{k-1}}.$$

Also $\mathbb E|S|\le p|B|$. Therefore

$$m_B(r)\le\frac{|B|}{p(1-p)^{4C_{k-1}}}.$$

Choosing $p=1/(4C_{k-1}+1)$ gives

$$m_B(r)\le\mathrm e(4C_{k-1}+1)|B|,$$

which proves the recurrence. If fewer than $k$ distances exist, the preceding-rank bound suffices instead.

Combined with layer support, this already proved divergence when every fixed layer prefix is $o(n)$, but yielded only a logarithmic quantitative bound in $n/h$. It is not a failed proof: it is a weaker valid argument. Lemma 2 supersedes it by proving $m_B(r)\le2k|B|$, and consequently a quadratic rather than exponential bound on the total number of pairs at the first $k$ distances.

D.2. Convex-subset and point-deletion transfers

Applying a rare-distance theorem to a convex subset $B\subset A$ controls $m_B(r)$, not $m_A(r)$. Restoring the other points may add both cross-pairs and pairs among the restored points. No bound was established that keeps enough of these distances below the full threshold $n$.

The small even-order argument does not justify a general deletion principle. It works because pair counting forces a unique diameter and exactly $n+1$ occurrences of every other class. Deleting one endpoint removes the diameter and cannot eliminate any other class: at most $n-1$ incident pairs disappear, leaving at least two occurrences. Outside this equality case, the needed residual classification and multiplicity control are missing.

D.3. Generic deformation and the strict half-width condition

Generic perturbation typically destroys equalities. The regular hexagon with its center has multiplicities $(12,6,3)$, hence $R=2$ at $n=7$; a generic small perturbation has 21 distinct distances, all rare. Thus a lower bound after perturbation gives no lower bound before perturbation. The usable affine hypothesis in Section 3 is preservation of every original equal-distance relation, not merely injectivity or preservation of inequalities between previously distinct lengths.

In the coordinate-splitting proof, the long-horizontal condition must be $a>W/2$, not $a\ge W/2$. The proof excludes neighbors on both abscissae $x-a$ and $x+a$ only in the strict case. Points on the middle fiber are therefore handled separately by globally rare vertical distances. The distinctness of these two collections uses the splitting hypothesis; it is not a conclusion of their being represented by different pairs.

D.4. Translation directions and graph excess

A fixed nonzero displacement vector gives a graph made of finite translation chains, so its unordered edge count is at most $n-1$. This suggested extracting rare distances from extremal displacement vectors. The missing step is that a Euclidean distance can combine many different directions of the same norm. Rarity of a direction class is not rarity of the full distance class. The affine hypothesis solves this only by restricting a distance to at most two unoriented directions of the form $(a,b)$ and $(a,-b)$.

Likewise, a rich distance graph has more edges than vertices and therefore a component of positive graph excess. Appendix C.2 forces a larger distance within such a component. This does not count distinct larger classes: many rich classes might lead to the same larger one. No injection or bound on this multiplicity of dependence was found. The abstract construction in Section 4 shows that even strong hereditary restrictions on maximum-color graphs cannot substitute for relations between different Euclidean lengths.

D.5. Selecting only the smallest and second-largest distances

The second-largest distance is not always rare. The supplied Vesztergombi paper states and constructs examples attaining $m_A(d_2)=3n/2$. Its original passage was read; the direct rank bound here does not assume the contrary.

Nor does selecting the smaller-multiplicity class among the shortest and second-largest distances suffice. The supplied Clemen–Dumitrescu–Liu Proposition 1.5 gives constructions with

$$m_A(d_2)\ge3m,\qquad m_A(\delta)\ge3n-5m+o(m).$$

Taking $m$ asymptotic to $3n/8$ gives both multiplicities at least $9n/8+o(n)$. The statement and construction passage were read during the investigation. This source construction was not independently reconstructed in full; it is recorded as a sourced obstruction and is not needed for any independently proved bound above.

D.6. Many rich distances do not give few rare distances

The supplied Bhowmick construction was read. In its odd-order case, two rotated regular $(m+1)$-gons sharing one vertex give $n=2m+1$ points. Each of $\lfloor m/2\rfloor$ selected polygon chord lengths has at least

$$2(m+1)=n+1$$

unordered occurrences, because the two polygons share no pair of vertices. This establishes many rich classes. It does not control the additional distances between the two polygons and does not bound the number of rare classes. The analogous reflection construction has the same logical limitation for the present problem. These constructions were not treated as counterexamples to either requested assertion.

Appendix E. Exact endpoint, verification status, and unresolved uncertainties

The range $n\ge5$ is retained. At four points, the coordinates

$$(0,0),\quad(1,0),\quad(1/2,\sqrt3/2),\quad(1/2,-\sqrt3/2)$$

give five unit pairs and one pair of length $\sqrt3$, hence $R=1$. This does not bear on the intended first question and does not obstruct the asymptotic question.

What is proved here. The depth-sum restriction, angular degeneracy lemma, layer and separation bounds, coordinate-splitting theorem, affine criterion, exact model counts, and abstract countermodel/nonrealizability calculation have explicit arguments in this record. The finite-order conclusion through ten points uses the stated published classifications as inputs. In particular, the nine-point models are identified by exhibiting four pairwise nonsimilar examples when the classification permits only four types, not by guessing figures that were not reproduced.

What is imported. The diameter theorem and the few-distance classifications are source theorems, not discoveries or exhaustive classifications independently reproved in this investigation. The twelve-point uniqueness input is Shinohara's Theorem 1.2(b). The optional $O(s^{4/3})$ incidence estimate is used only as stated in the supplied Clemen–Dumitrescu–Liu Section 1.4; the independent elementary bounds do not depend on it. A transcription coefficient in their two-layer proof is corrected explicitly in Section 5. No claimed error in the final source theorem follows from that correction.

Checks performed. Exact coordinate calculations were checked by squared-distance formulas, integer distance-degree tables, and the symbolic three-orbit table in Appendix A; their pair totals are respectively 36, 66, and 55 for nine, twelve, and eleven points. The checks are reproducible from the tables without floating-point tests. The delicate geometric checks include nonvertex convex combinations on degenerate layers, strict increases of distance, leftmost-coordinate ties in the closed semicircle, cosine monotonicity through the endpoint $\pi$, existence of enough ranked distances, and survival of nondiameter classes after endpoint deletion.

Unresolved finite cases. No argument excludes every eleven-point five-distance set with only one rare class. Nor does the twelve-point uniqueness theorem classify arbitrary eleven-point five-distance sets. At twelve points the only remaining one-rare pattern identified here is five multiplicities 13 and a unique diameter. Appendix A.3 gives further necessary conditions, not a proof of nonexistence. The thirteen-point restrictions are likewise only necessary conditions.

Unresolved asymptotic case. A bounded-$R$ sequence must have a macroscopic convex layer at bounded depth. No argument controls this regime. For convex sets, the available distinct-distance and multiplicity counts reduce bounded $R$ to $q=n/2+O(1)$, but no general bounded-excess classification or stability theorem was proved or supplied. Even divergence restricted to arbitrary convex sets is not established by this record.

Scope of the conclusion. Neither universal assertion has been proved or disproved here. The abstract colorings are explicitly non-Euclidean and are not planar counterexamples. There is no claimed extension from convex sets, affine-deformable sets, or sparse-layer sequences to all planar sets. No statement about novelty or the complete later literature was verified. No unavailable earlier research notes, external web material, or later papers were consulted. This record does not assert that the remaining gaps have been resolved.