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Math-6138-A2 · revision r1 · on Math-6138

gpt pro 5.2 attempt on Math-6138

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Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Erdős Problem 1084 (unit distances under separation)
  2. FORMAL RESTATEMENT
  3. QUICK LITERATURE/CONTEXT CHECK
  4. ATTACK PLAN
  5. WORK
  6. Fast reality check.
  7. VERIFICATION
  8. FINAL

Erdős Problem 1084 (unit distances under separation)

FORMAL RESTATEMENT

For an integer $d\ge 1$ and $n\in\mathbb{N}$, let $f_d(n)$ be the minimum integer such that for every set $P\subset\mathbb{R}^d$ of size $|P|=n$ with $\|x-y\|\ge 1$ for all distinct $x,y\in P$, the number of unordered pairs $\{x,y\}\subset P$ with $\|x-y\|=1$ is at most $f_d(n)$. The problem asks for estimates of $f_d(n)$ (as $n\to\infty$ with $d$ fixed).

QUICK LITERATURE/CONTEXT CHECK

The problem statement records: $f_1(n)=n-1$, $f_2(n)<3n$ by a degree argument, and a refined planar bound $f_2(n)<3n-c n^{1/2}$ due to Erdős. It also records a conjectural exact planar value for certain $n$, and a claimed asymptotic refinement in $d=3$. I do not add external results here.

ATTACK PLAN

  • Proof track (partial). Relate $f_d(n)$ to the kissing number in $\mathbb{R}^d$ to get a clean linear upper bound. In $d=2$, exhibit a triangular-lattice construction giving $3n-O(\sqrt n)$ unit pairs and compute its exact unit-pair count.
  • Disproof track. Not applicable (estimation problem).

WORK

Fast reality check.
For $d=2$, the configuration of one point and a regular hexagon around it gives $n=7$ and $12$ unit pairs. A brute-force check on the triangular lattice within radius $2$ shows this is best possible among 7-point subsets of that lattice.
Definition (Kissing number constant): Let $\tau_d$ be the maximum size of a set $S\subset \mathbb{S}^{d-1}$ with $\|u-v\|\ge 1$ for all distinct $u,v\in S$. Equivalently, $\tau_d$ is the maximum number of points on the unit sphere such that mutual distances are at least $1$. (Geometrically, $\tau_d$ equals the maximum number of disjoint radius-$1/2$ balls that can simultaneously touch a central radius-$1/2$ ball.)
Lemma (Degree bound via kissing number): Let $P\subset\mathbb{R}^d$ have all pairwise distances $\ge 1$. Form the graph $G$ on vertex set $P$ with an edge between $x\ne y$ iff $\|x-y\|=1$. Then every vertex has degree at most $\tau_d$, and hence \[ \#E(G)\le \frac{\tau_d}{2}\,|P|. \] In particular, $f_d(n)\le \frac{\tau_d}{2}n$ for all $n$.
Proof: Fix $p\in P$. Let $N(p)=\{q\in P: \|q-p\|=1\}$ be its unit neighbors. For any distinct $q,r\in N(p)$, we have $\|q-r\|\ge 1$ by the separation assumption. Map each $q\in N(p)$ to the unit vector $u_q=(q-p)/\|q-p\|\in\mathbb{S}^{d-1}$. Then $\|u_q-u_r\|=\|q-r\|$ (since $\|q-p\|=\|r-p\|=1$), so $\|u_q-u_r\|\ge 1$. Thus $\{u_q: q\in N(p)\}\subset\mathbb{S}^{d-1}$ is a set of size $|N(p)|$ with pairwise distances at least $1$, so $|N(p)|\le \tau_d$ by definition. The edge bound follows by summing degrees and dividing by $2$. ∎
Lemma (Planar triangular-lattice construction): Let $m\ge 0$. Consider the triangular lattice in axial coordinates $\Lambda=\{(a,b): a,b\in\mathbb{Z}\}$ with unit-distance edges between points whose differences are $(\pm1,0),(0,\pm1),(\pm1,\mp1)$. Let \[ H_m:=\{(a,b)\in\Lambda: \max(|a|,|b|,|a+b|)\le m\} \] (the "centered hexagon" of radius $m$). Then \[ |H_m|=3m^2+3m+1,\qquad \#\{\{x,y\}\subset H_m: \|x-y\|=1\}=9m^2+3m. \] In particular, this gives a configuration with $n=3m^2+3m+1$ points and $3n-(6m+3)=3n-O(\sqrt n)$ unit pairs.
Proof: The count $|H_m|=3m^2+3m+1$ is standard for a centered hexagon: there is a central point and $6k$ points on the "ring" at lattice distance $k$ for each $1\le k\le m$, so $|H_m|=1+\sum_{k=1}^m 6k=1+3m(m+1)=3m^2+3m+1$.

Now count unit edges in the induced unit-distance graph on $H_m$. Every interior vertex has degree $6$. The boundary of $H_m$ is a hexagon with $6m$ boundary vertices: it has $6$ corner vertices and $6(m-1)$ non-corner boundary vertices. A direct neighbor check in axial coordinates shows:

  • each corner boundary vertex has exactly $3$ neighbors within $H_m$,
  • each non-corner boundary vertex has exactly $4$ neighbors within $H_m$.
Thus the total degree deficit from $6$ is \[ \mathrm{def}:=\sum_{v\in H_m} (6-\deg(v)) = 6\cdot(6-3) + 6(m-1)\cdot(6-4) =18+12(m-1)=12m+6. \] Therefore \[ 2|E|=\sum_{v\in H_m}\deg(v)=6|H_m|-\mathrm{def}. \] Plugging $|H_m|=3m^2+3m+1$ and $\mathrm{def}=12m+6$ gives \[ 2|E|=6(3m^2+3m+1)-(12m+6)=18m^2+18m+6-12m-6=18m^2+6m, \] so $|E|=9m^2+3m$.

Finally, since $m\asymp \sqrt n$ when $n=3m^2+3m+1$, the error term $6m+3$ is $O(\sqrt n)$, yielding $|E|=3n-O(\sqrt n)$. ∎

VERIFICATION

  • The kissing-number lemma is purely geometric and uses only the separation hypothesis.
  • In the planar construction, I checked the edge counts computationally for $m=0,1,2,3,4$: $(|H_m|,|E|)=(1,0),(7,12),(19,42),(37,90),(61,156)$, matching $3m^2+3m+1$ and $9m^2+3m$.
  • Note: the problem statement contains a conjectural exact formula $f_2(3n^2+3n+1)=9n^2+6n$. The natural centered-hexagon triangular-lattice patch above gives $9n^2+3n$ unit pairs for that number of points; I did not resolve the discrepancy.

FINAL

UNRESOLVED.
  1. (i) Strongest proved partial result here: for all $d$, $f_d(n)\le (\tau_d/2)n$; and in $d=2$ there exist configurations with $3n-O(\sqrt n)$ unit pairs.
  2. (ii) First gap: determine the correct second-order term in $d=2$ (and in particular the exact value at the "hexagonal numbers"), and in $d=3$ prove or disprove the claimed $6n-\Theta(n^{2/3})$ behaviour.
  3. (iii) Top 3 next moves: $1$ for $d=2$, formulate the extremal problem as maximizing edges in a planar unit-distance graph with minimum separation and analyze boundary effects rigorously; $2$ attempt a discrete isoperimetric inequality on the hexagonal/triangular lattice to show optimality of near-hexagonal patches; $3$ computationally optimize for small $n$ to test the conjectured exact values.
  4. (iv) Minimal counterexample structure (to "triangular lattice is extremal" in $d=2$): a planar packing close to hexagonal but with local defects that increase the unit-edge count without violating separation, likely concentrated near the boundary.