Erdős (Erdos) Problem #706: status in 2026

Erdős Problem #706

Let L(r) be such that if G is a graph formed by taking a finite set of points P in ℝ^2 and some set A⊂ (0,∞) of size r, where the vertex set is P and there is an edge between two points if and only if their distance is a member of A, then χ(G)≤ L(r). Estimate L(r).

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-706,
  title        = {Erdős Problem #706},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-706}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Let $L (r)$ be such that if $G$ is a graph formed by taking a finite set of points $P$ in $R^{2}$ and some set $A \subset (0, \infty)$ of size $r$ , where the vertex set is $P$ and there is an edge between two points if and only if their distance is a member of $A$ , then $χ (G) \leq L (r)$ .

Estimate $L (r)$ . In particular, is it true that $L (r) \leq r^{O (1)}$ ?

The case $r = 1$ is the Hadwiger-Nelson problem, for which it is known that $5 \leq L (1) \leq 7$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«706».

theorem erdos_706 :
    ∃ C : ℕ, ∀ᶠ r : ℕ in atTop, ∀ (P : Finset ℝ²) (A : Finset ℝ), A.card = r →
      (∀ a ∈ A, 0 < a) →
      (SimpleGraph.DistancePlaneGraph (A : Set ℝ) (P : Set ℝ²)).chromaticNumber ≤
        (r ^ C : ℕ)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/706. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/706
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.