Beyond Brown–Gerver–Ramsey
Only unit steps.
Few points on a line.
Walk forever, one positive coordinate step at a time. In three, four and six dimensions, these constructions put a small bound on every line.
Brown’s planar walks force arbitrarily large collinear sets. Gerver and Ramsey found bounded collinearity in 3D. Here, positive basis walks reach sharper bounds in small dimensions. Finite-step companion ↗
| 3D | 4D | 5D* | 6D |
|---|---|---|---|
| <7 | <4 | <4 | <3 |
*5D embeds the 4D walk; one direction is unused.
How is this projected from higher dimensions?
For P = (x₀, …, xd−1), the 3D display point is Q = Σ xjvj. These are the actual vectors vj used:
| Direction | X | Y | Z |
|---|
Optional compression shortens the component along the prefix’s endpoint direction u: Q′ = Q − (1 − 1/c)(Q · u)u. It changes lengths and angles, not the underlying walk. We then rotate about a fixed 3D center and drop depth. A fixed bounding sphere sets one uniform scale: rotating does not re-fit or re-center the walk. Apparent shortening as a segment turns away is ordinary foreshortening.
The initial camera shows the broad face of the walk. Projection can create apparent alignments: the picture is not a collinearity test. The 6D map is illustrative, not the manuscript’s five-real-dimensional auxiliary space. Claims concern the original integer coordinates.
The key idea
Count letters. Prevent alignment.
A letter j means “take step ej.” The vertex is just the word’s running letter count:
Three vertices a < b < c align exactly when the intervening blocks have the same letter proportions:
Choose a word that prevents these repetitions. For k collinear vertices, all k − 1 block ratios must agree, even with unequal block lengths.
Where the bounds come from
6D: a binary-state recurrence and a 2-adic identity exclude collinear triples. 4D: ternary-state returns give four step types; a modulo-3 contradiction excludes four collinear vertices. 3D: subdividing the diagonal return steps gives two sets, each meeting a line at most three times—six in total. The updated manuscript exhibits a line containing six vertices of this 3D construction and three vertices on one line in its 4D construction, without claiming optimality among all walks.
The open cases in this snapshot include no-three in 4D/5D and no-four in 3D. The manuscript also records a separate five-direction 5D candidate checked for 600,000 terms, but not proved. The 5D example on this site remains only the proven 4D embedding.
The constructor
A short rule. An infinite walk.
Run to see the first 13 vertices.
Runs locally in a disposable Python worker; first run downloads Pyodide. 120-second limit. Don’t paste secrets or untrusted code. Finite checks are not infinite proofs.
How the symbols become steps.
Only the final digit strings are step words: each digit j instructs the walk to add ej. Each sample shows the first 32 symbols of its sequence.
- Intermediate source word · 3D–5D only
- The substitution produces letters A–E. These letters organize the construction; they are not directions yet.
ABAACAABABADEABABAACAABAACAABAAC - 3D coding → step word
- A→0 · B→1 · C→20 · D/E→21
01002000101021210101002000100200 - 4D / embedded 5D coding → step word
- A→0 · B→1 · C→2 · D/E→3
01002001010330101002001002001002 - 6D state transitions → step word
- There is no A–E source word. Each adjacent state pair (σn, σn+1) selects j: 01→0, 12→1, 23→2, 30→3, 02/13→4, 20/31→5.
04301504342304301501251504301504
The 5D example uses the 4D step word unchanged, so e4 is never selected. The 6D digits use the site’s fixed coordinate order.
Finite verification
100,000 steps checked in every dimension.
Each exact scan checks 100,001 vertices and 5,000,050,000 vertex pairs in the original integer coordinates. All four completed with no forbidden collinear set found.
These finite-prefix passes are reproducible evidence, not infinite proofs, optimality claims, independent human review or formal verification.
Paper & credit
Stijn Cambie, Erik Kalviainen & Jeffrey Shallit
Brown-Gerver-Ramsey Theorems in Small Dimensions. arXiv:2609.20366.
Scope & software attribution
The stated infinite bounds are the manuscript’s results, not conclusions from these pictures or finite scans. No optimality, independent human review or formal verification is claimed. Software maintained by Erik Kalviainen with AI assistance; paper authorship does not imply review of this software. The submitted manuscript is distributed under the Creative Commons Attribution 4.0 license.