# P100007: Chromatic number of the integer distance graph on cubes

- ID: `P100007`
- Reference: `chromatic-number-cube-distance-graph`
- Page: https://theoremdb.org/statements/P100007
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/chromatic-number-cube-distance-graph?representation=markdown
- Record maturity: Archived record

## The problem

Let \(D = \{k^3 : k \ge 1\}\) and let \(G(\mathbb{Z}, D)\) be the graph on the integers in which \(x\) and \(y\) are adjacent exactly when \(|x - y| \in D\). Is \(\chi(G(\mathbb{Z}, D)) = 3\)?

### Problem setup

- **Remark.** The distance graph G(Z, D) has the integers as vertices, with x adjacent to y exactly when |x - y| lies in D.
- **Definition.** Here D is the set of positive perfect cubes 1, 8, 27, 64, and so on.

### What counts as a solution

- Exhibit a proper 3-colouring of the integers avoiding all cube differences, together with a proof that 2 colours do not suffice; or prove that more than 3 colours are required.

## Status

**Resolution status:** Solution submitted.
**Lean verification:** Not Lean-verified.

Every finite coloring of the integers contains a same-colored pair at a positive cube distance, so the graph has chromatic number \(\aleph_0\). [1](#reference-1)

**Recorded scope.** the integer distance graph whose allowed distances are all positive cubes

[Read this exact record and its supporting evidence](/records/cdg-claim-countably-infinite-chromatic-number)


### Resolution argument

Let \(D=\{n^3:n\geq1\}\). Bergelson and Leibman's polynomial van der Waerden theorem, applied to the zero polynomial and \(p(n)=n^3\), gives two same-colored integers \(x\) and \(x+n^3\) in every finite coloring of \(\mathbb Z\). Those vertices are adjacent in \(G(\mathbb Z,D)\), so no finite coloring is proper. The graph is countable, which supplies the upper bound \(\aleph_0\).

This refutes the candidate value \(3\). Guerreiro, Ruzsa, and Silva describe the same fact by saying that the positive values of \(p\) form a chromatically intersective set, or a ladder. The square-distance and triangular-distance cases follow from the same polynomial theorem.

## Research packet

### Saved solution and evidence

**Proposition 1 (The cube-distance graph has countably infinite chromatic number).** Every finite coloring of the integers contains a same-colored pair at a positive cube distance, so the graph has chromatic number \(\aleph_0\).

Let \(D=\{n^3:n\geq1\}\). Bergelson and Leibman's polynomial van der Waerden theorem, applied to the zero polynomial and \(p(n)=n^3\), gives two same-colored integers \(x\) and \(x+n^3\) in every finite coloring of \(\mathbb Z\). Those vertices are adjacent in \(G(\mathbb Z,D)\), so no finite coloring is proper. The graph is countable, which supplies the upper bound \(\aleph_0\).

This refutes the candidate value \(3\). Guerreiro, Ruzsa, and Silva describe the same fact by saying that the positive values of \(p\) form a chromatically intersective set, or a ladder. The square-distance and triangular-distance cases follow from the same polynomial theorem.

### Background and intake notes

Screened together with squares, triangular numbers, primes, palindromes and powers of two. Palindromic distances force at least 10 colours because 1 through 9 are all palindromes, so that variant is degenerate. Square and triangular distances produced chromatic numbers that grew with the window, which is a different and probably harder question.

- Original intake status: Status not established. Chromatic numbers of integer distance graphs are a studied topic and this particular case may well be settled in the literature; no search was performed. Treat the value 3 as a computational conjecture on finite windows, not as a claim about the infinite graph.
- No literature search was carried out. The prime-distance case is known to have chromatic number 4, which suggests this family has been worked over and that the cube case may be known.
- Finite windows give only lower bounds for the infinite graph. A complete answer needs a periodic colouring for the upper bound.

- Recorded example: Colouring n by n mod 3 fails, since 64 is a cube and 64 is congruent to 1 modulo 3.

### Other known results

- **Proposition 2** (supported): Positive values of an integer-valued polynomial with zero constant term form a ladder, the distance-set condition equivalent to infinite chromatic number. [2](#reference-2)

### Prior approaches

- **Route 1** (incomplete method): Every periodic coloring fails immediately because a period \(q\) is also crossed by the allowed distance \(q^3\). [2](#reference-2)

### Runnable artifacts

- **Artifact 1** (executable material): The induced graphs on widths 20, 30, 40, and 60 each have chromatic number 3.

### Computational notes

- Exact chromatic number of the induced subgraph on windows of width 20, 30, 40 and 60 was 3 in every case. For comparison, square distances gave 3, 4, 4, 5 and triangular distances gave 3, 3, 4, 5 on the same windows.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `chromatic-number-cube-distance-graph`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## Lean verification

A solution has been submitted and awaits independent review.

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## References

2 entries have incomplete source metadata. Each affected row names the fields that still need editorial review.

1. <a id="reference-1"></a>Bergelson and Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, Theorem C and Corollary 1.11 https://people.math.osu.edu/leibman.1/preprints/psz.pdf
   - website; reference source; checked 2026-07-24
   - Source metadata incomplete: publication-style citation.
   - Source use: citation_only
   - Source named by the research packet.
2. <a id="reference-2"></a>Guerreiro, Ruzsa, and Silva, Monochromatic Paths for the Integers, Corollary 4 and section 5 https://arxiv.org/abs/1606.00418
   - Also cited at The general obstruction follows from Corollary 4; the periodic obstruction is elementary
   - preprint; reference source; checked 2026-07-24
   - Source metadata incomplete: publication-style citation, source version, relevance note.
   - Source use: citation_only