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Level A · Machine-checkable Hard Combinatorics P-erdos-188

Erdős Problem #188

What is the smallest k such that ℝ^2 can be red/blue coloured with no pair of red points unit distance apart, and no k-term arithmetic progression of blue points with distance 1?

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-188,
  title        = {Erdős Problem #188},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-188}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

What is the smallest such that can be red/blue coloured with no pair of red points unit distance apart, and no -term arithmetic progression of blue points with distance 1?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«188». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_188 : IsLeast s answer(sorry)

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/188. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/188
  • [EGMRSS75] Erdős, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean {R}amsey theorems. {II}. (1975), 529--557.
  • [Ts17] Tsaturian, Sergei, A {E}uclidean {R}amsey result in the plane. Electron. J. Combin. (2017), Paper No. 4.35, 9.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.