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Math-910-A1 · revision r1 · on Math-910

Reused writeup on Math-910 (gpt pro 5.2)

No claim recorded · LLM-generated · gpt pro 5.2 ·

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 704 --- collection record
  2. PROVENANCE AND STATUS
  3. REUSED SOURCE 1
  4. Erdős Problem #704
  5. FORMAL RESTATEMENT
  6. QUICK LITERATURE/CONTEXT CHECK
  7. ATTACK PLAN
  8. WORK
  9. VERIFICATION
  10. FINAL
  11. COMPLETION ESTIMATE

Erdős Problem 704 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/704

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/704.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/704.tex.

Erdős Problem #704

FORMAL RESTATEMENT

Let $G_n$ be the (infinite) graph with vertex set $\mathbb{R}^n$ and an edge between two distinct vertices $x,y\in\mathbb{R}^n$ if and only if $\|x-y\|_2=1$. Let $\chi(G_n)$ denote its (ordinary) chromatic number. Tasks:
  1. Give upper and lower bounds on $\chi(G_n)$ as a function of $n$.
  2. Decide whether $\chi(G_n)$ grows exponentially in $n$.
  3. Decide whether $\lim_{n\to\infty} \chi(G_n)^{1/n}$ exists.

QUICK LITERATURE/CONTEXT CHECK

The problem statement records an exponential lower bound due to Frankl--Wilson and an exponential upper bound due to Larman--Rogers (with an alternative proof by Prosanov). The corresponding discussion pages and papers exist, but I do not reproduce those deep arguments here; I give elementary bounds and sanity checks only.

ATTACK PLAN

  • Lower bounds: build explicit finite unit-distance subgraphs with large chromatic number. The easiest is a clique (regular simplex) giving $\chi(G_n)\ge n+1$.
  • Upper bounds: partition $\mathbb{R}^n$ into small-diameter cells (cubes) and color the cells periodically so that same-colored cells are separated by distance $>1$.

WORK

Lemma 704.1 (clique lower bound via a regular simplex). For every $n\ge 1$, the unit distance graph $G_n$ contains a clique of size $n+1$. Consequently \[\chi(G_n)\ge n+1. \]

Proof. In $\mathbb{R}^n$ there exists a regular simplex with $n+1$ vertices, i.e. points $v_0,\dots,v_n$ with all pairwise distances equal to some constant $r>0$. (One explicit construction is to take the $n+1$ vectors in $\mathbb{R}^{n+1}$ given by $e_i-\tfrac{1}{n+1}\sum_{j=0}^n e_j$ and view them in the $n$-dimensional subspace of vectors summing to $0$.) Scaling the simplex by factor $1/r$ yields $n+1$ points in $\mathbb{R}^n$ with all pairwise distances exactly $1$. These vertices form a clique $K_{n+1}$ in $G_n$, and any proper coloring needs at least $n+1$ colors. $\square$

Lemma 704.2 (elementary cube-tiling upper bound). For every $n\ge 1$, \[\chi(G_n)\le \left(\lceil\sqrt{n}\rceil+3\right)^n. \] In particular, $\chi(G_n)\le (\sqrt{n}+3)^n$.

Proof. Let $s:=1/(\lceil\sqrt{n}\rceil+1)$. Tile $\mathbb{R}^n$ by axis-parallel cubes of side length $s$ whose vertices lie in the lattice $s\mathbb{Z}^n$. Any two points within the same cube have Euclidean distance at most the cube diameter $s\sqrt{n}$. Since $\lceil\sqrt{n}\rceil\ge \sqrt{n}$, we have \[s\sqrt{n}=\frac{\sqrt{n}}{\lceil\sqrt{n}\rceil+1}\le \frac{\sqrt{n}}{\sqrt{n}+1}<1. \] Thus no two points at distance exactly $1$ can lie in the same cube.

Now color the cubes periodically by their lattice indices modulo $t:=\lceil 1/s\rceil+1$ in each coordinate. Since $1/s=\lceil\sqrt{n}\rceil+1$, we have $t=\lceil\sqrt{n}\rceil+2$. This gives at most $t^n=(\lceil\sqrt{n}\rceil+2)^n$ colors. Finally, to ensure cubes of the same color are separated by distance $>1$, observe that if two cubes have the same color then their indices differ by a multiple of $t$ in every coordinate, so in at least one coordinate the index differs by at least $t$ unless they coincide. In that coordinate, the distance between the corresponding cube intervals is at least $(t-1)s$. But \[(t-1)s=(\lceil\sqrt{n}\rceil+1)\cdot \frac{1}{\lceil\sqrt{n}\rceil+1}=1. \] To avoid the boundary case "$=1$", we can instead take $t:=\lceil\sqrt{n}\rceil+3$, which yields $(t-1)s>1$ and still uses only $(\lceil\sqrt{n}\rceil+3)^n$ colors. Then any two points in cubes of the same color are at distance $>1$, so they are non-adjacent, and the induced vertex-coloring of $\mathbb{R}^n$ is proper. Hence $\chi(G_n)\le (\lceil\sqrt{n}\rceil+3)^n$. $\square$

FAST REALITY CHECK.

  • $n=1$: Lemma 704.1 gives $\chi(G_1)\ge 2$. A 2-coloring is obtained by coloring $x\in\mathbb{R}$ by the parity of $\lfloor x\rfloor$; any two reals at distance $1$ have floors differing by $1$, hence different colors. So $\chi(G_1)=2$.
  • $n=2$: Lemma 704.1 gives $\chi(G_2)\ge 3$ and Lemma 704.2 gives $\chi(G_2)\le (\lceil\sqrt{2}\rceil+3)^2=25$ (very weak).

VERIFICATION

  • Lemma 704.1: verified that scaling preserves unit distances and that a clique forces at least that many colors.
  • Lemma 704.2: the only delicate point is ensuring same-colored cubes are at distance strictly greater than $1$; I addressed this by padding the period from $t=\lceil\sqrt{n}\rceil+2$ to $t=\lceil\sqrt{n}\rceil+3$.

FINAL

UNRESOLVED.
  1. (i) Strongest proved partial results here:
    • Elementary bounds: $n+1\le \chi(G_n)\le (\lceil\sqrt{n}\rceil+3)^n$ (Lemmas 704.1--704.2).
    • The problem text records much stronger exponential bounds (Frankl--Wilson lower bound and Larman--Rogers/Prosanov upper bound), but I did not reproduce those proofs.
  2. (ii) First gap: establish sharp exponential growth rates (or the existence of $\lim_{n\to\infty}\chi(G_n)^{1/n}$) with proofs that do not rely on deep external theorems not included in the statement.
  3. (iii) Top 3 next moves:
    1. Work out an explicit finite unit-distance graph in $\mathbb{R}^n$ with chromatic number $(c+o(1))^n$ (re-deriving a Frankl--Wilson-type lower bound from first principles).
    2. Improve the cube-tiling argument by using better sphere/ball packings or Voronoi cells to reduce the base of the exponential upper bound.
    3. Numerically compute/approximate $\chi(G_n)$ for small $n$ in restricted models (e.g. measurable colorings or periodic colorings) to guess limiting behavior.
  4. (iv) Minimal counterexample structure (to existence of the limit): one would need oscillation between lower-bound constructions and upper-bound constructions so that $\chi(G_n)^{1/n}$ fails to converge; this would likely involve different extremal configurations dominating in different dimensions.

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.