4) V4 For every integer m≥1, there is a lattice set S in [29m,31m]×[19m,21m] with 10⁹|S|≥(2m+1)² such that F∪S has diameter at most 60√2m and |area(a,p,q)−100m|>90m for all a∈F and p,q∈F∪S, where F={(0,0),(60m,60m),(60m,0)}.
kernel-checked, filed Tue Sep 08 2026 02:24:32 GMT+0000 (Coordinated Universal Time) by @coleski
These sets give quadratic cliques in the three-anchor compatibility relaxation for Erdős #352.
Quadratic cliques in the three-anchor compatibility relaxation.
Proved in Lean and kernel-verified by Jig. The written proof is given below. This is a limitation of a relaxation, not a solution or counterexample to Erdős #352.
Statement
For every integer m≥1, put N=100m, A=(0,0), B=(60m,60m), H=(60m,0), F={A,B,H}, and Q=([29m,31m]×[19m,21m])∩Z². There exists S⊂Q satisfying 10⁹|S|≥(2m+1)².
All distances in F∪S are at most D=60√2m. Every triangle with vertices in F∪S and at least one vertex in F satisfies |area(triangle)−N|>90m>D.
Repeated vertices are permitted. In particular S is a clique in Kitamura's high-side compatibility branch with diameter pair A,B and high anchor H. Its size is at least N²/2,500,000,000,000. Thus the maximum branch clique number, and hence the maximum branch chromatic number, cannot be O(N).
Proof
All distances in F∪Q are at most D. The maximum squared distances from Q to A and H are 1402m², from Q to B are 2642m², and within Q are 8m²; distances between anchors are at most 7200m²=D². Also D<90m<N.
The area of ABH is 1800m². For q∈Q, the areas of ABq, AHq and BHq are at least 240m², 570m² and 870m², respectively. Each exceeds 190m=N+90m. Thus every distinct-vertex triangle with at least two anchors already satisfies the claimed gap. Every repeated-vertex triangle has area zero, and its gap is N>90m.
It remains to choose S so that all triangles with exactly one anchor pass. For each q=(x,y)∈Q assign three positive coordinate pairs:
At A: (u,v)=(x,y); at B: (u,v)=(60m−x,60m−y); at H: (u,v)=(60m−x,y).
These are translations followed, when necessary, by coordinate reflections, so absolute determinants are preserved. At each anchor, N/4≤u≤N. Color q at that anchor by floor(10Nv/u) modulo 1000.
For two points p,q of the same color, write t=v/u, t'=v'/u', and k=floor(10Nt), k'=floor(10Nt'). The corresponding triangle area is uu'|t−t'|/2. If k=k', then |t−t'|<1/(10N), so its area is less than N/20=5m. If k≠k', congruence gives |k−k'|≥1000. The definition of floor then gives |t−t'|>999/(10N). Its area therefore exceeds (N/4)²·999/(10N)/2 = 999N/320 > 3N.
In either case the area lies outside [N−90m,N+90m]=[10m,190m].
The combined three-anchor color takes at most 1000³=10⁹ values. Since |Q|=(2m+1)², the pigeonhole principle gives a color class S of the asserted size. The preceding argument handles every one-anchor triangle; the distance and remaining triangle checks were already established. This proves the statement for every m≥1.
To identify the graph precisely, Kitamura's single_reject_reason checks candidate distances and triangles with two forced anchors (including repetitions); pair_conflict checks pair distances, repetitions, and triangles with one forced anchor. Our stronger uniform gap implies each of these checks passes. B is a normalized allowed diameter vector, H is high, and the branch loop considers every high H. Thus S is a clique in that actual branch, not merely in a different relaxed model.
Significance and provenance.
Kitamura explicitly describes the graph as a relaxation, not as an exact model for avoiding sets. The result here gives an asymptotic obstruction to obtaining a linear global bound by optimizing that relaxation's clique or coloring bounds alone. It does not contradict the valid computations for N=1,…,7 or Koizumi's reduction. Triples of three points of S remain unconstrained. Additional triple constraints or other pruning are not ruled out.
Developed during coleski's investigation on Jig with Codex assistance. The novelty check found no matching obstruction in the repository, the current #352 discussion, Jig #132, or the related sources examined. This is a bounded literature check, not a claim of established historical priority. Related coloring constructions are known; the contribution claimed here is this explicit obstruction for the three-anchor relaxation.
Source of the relaxation: https://github.com/KitaKen1/erdos352-koizumi-fn-upper-bound (exact_bound.py: single_reject_reason, pair_conflict, case_report). Discussion and discrete formulation: https://www.erdosproblems.com/forum/thread/352 (Koizumi, 24 December 2025; Kitamura, 3 June 2026).
Scope. For every integer m≥1, a finite lattice subset of [29m,31m]×[19m,21m] satisfying the stated cardinality, diameter, and three-anchor triangle-gap inequalities.