gpt pro 5.2 attempt on Math-671

FORMAL RESTATEMENT. Let $A\subset\mathbb{R}^2$ be the vertex set of a convex $n$-gon (i.e. $|A|=n$ and the points are in convex position). Let \[ U_\mathrm{conv}(A):=|\{\{x,y\}\subset A: |x-y|=1\}| \] be the number of unit-distance pairs among the vertices. Conjecture (Erdős--Moser): there exists an absolute constant $C$ such that for all $n$ and all convex $n$-gons, \[ U_\mathrm{conv}(A)\le Cn. \]

QUICK LITERATURE/CONTEXT CHECK. The problem statement reports known bounds: $O(n\log n)$ and more precise variants, and a lower bound construction with $2n-7$ unit distances. I do not re-prove those results here. Instead I record an elementary linear lower bound and an elementary $O(n^{3/2})$ upper bound (which holds without convexity).

ATTACK PLAN.

• Lower bound sanity: show that $\Omega(n)$ unit distances are attainable even with convexity (regular polygon).
• Upper bound (weak): apply the general unit-distance $O(n^{3/2})$ bound (Lemma 90.2) to convex point sets.
• (Unresolved) improvement: exploit convex position to beat the generic $n^{3/2}$ bound, aiming toward $O(n\log n)$ and then $O(n)$.

WORK.

Lemma 96.1 (convex linear lower bound). For every $n\ge 3$ there exists a convex $n$-gon with at least $n$ unit-distance pairs. Proof. Take a regular $n$-gon and scale it so that its side length is $1$. Then each of the $n$ sides is a unit distance. We claim no diagonal has length $1$. Indeed, in a regular $n$-gon with circumradius $R$, the chord length between vertices at step $k$ is $2R\sin(k\pi/n)$. After scaling to make the $k=1$ chord equal to $1$, we have $2R\sin(\pi/n)=1$. If $2R\sin(k\pi/n)=1$ also held for some $2\le k\le n-2$, then $\sin(k\pi/n)=\sin(\pi/n)$, which for $0<k\pi/n<\pi$ forces $k=1$ or $k=n-1$. Thus only adjacent pairs are at distance $1$, giving exactly $n$ unit distances. $\square$

Lemma 96.2 (generic $O(n^{3/2})$ upper bound). For every convex $n$-gon vertex set $A\subset\mathbb{R}^2$, \[ U_\mathrm{conv}(A)\le \frac{\sqrt 2}{2}\,n^{3/2}+\frac14 n. \] Proof. This is immediate from Lemma 90.2, which bounds the number of unit-distance pairs in any $n$-point set in the plane; convexity is an additional constraint and cannot increase the maximum. $\square$

FAST REALITY CHECK (regular polygons). For regular $n$-gons scaled to side length $1$, a direct computation confirms the unit-distance count equals $n$ for $3\le n\le 10$:

n=3 -> 3 unit pairs
n=4 -> 4 unit pairs
n=5 -> 5 unit pairs
...
n=10 -> 10 unit pairs

VERIFICATION.

• Lemma 96.1: verified the chord-length formula and the fact $\sin(k\pi/n)=\sin(\pi/n)$ has only the symmetric solutions $k=1,n-1$ in $\{1,\dots,n-1\}$.
• Lemma 96.2: relies only on Lemma 90.2, whose proof uses the circle-intersection ($\le2$) property and a $K_{2,3}$ extremal bound.
• Boundary case $n=3$: the regular triangle has 3 unit sides, consistent with Lemma 96.1.

FINAL. UNRESOLVED

1. (i) Strongest proved partial result. There are convex $n$-gons with $\ge n$ unit distances (Lemma 96.1). Conversely, any convex $n$-gon has $O(n^{3/2})$ unit distances by the general bound (Lemma 96.2).
2. (ii) First gap (crisp). Use convexity to improve the $O(n^{3/2})$ bound, ideally proving $U_\mathrm{conv}(A)=O(n)$ for all convex $n$-gons.
3. (iii) Top 3 next moves.
   • Prove a per-vertex bound: show that for a convex $n$-gon, each vertex can be at unit distance from at most $O(\log n)$ (or $O(1)$) other vertices, then sum over vertices.
   • Translate unit distances into incidences between vertices and unit circles, and exploit the cyclic order on the convex hull to strengthen the $K_{2,3}$-free extremal-graph estimate.
   • Search computationally for convex $n$-gons with many unit distances to guess extremal patterns (e.g. mixtures of arcs/chords), which may suggest the right combinatorial decomposition.
4. (iv) Likely structure of a minimal counterexample. A counterexample to linearity would require a convex $n$-gon with superlinear many unit chords. Such a configuration would likely place many vertices on a few arcs of circles of radius 1 around many other vertices, creating a highly constrained "almost-circular" structure with many repeated chord lengths.