Erdős Problem #159 — open problem in graph theory

Erdős Problem #159

There exists some constant c>0 such that R(C_4,K_n) ≪ n^2-c. The prize of 100 is offered in [Er78] for a proof or disproof. This problem is #17 in Ramsey Theory in the graphs problem collection.

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-159,
  title        = {Erdős Problem #159},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-159}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

There exists some constant $c > 0$ such that $R (C_{4}, K_{n}) ≪ n^{2 - c} .$

The prize of $100 is offered in [Er78] for a proof or disproof.

This problem is #17 in Ramsey Theory in the graphs problem collection.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«159».

theorem erdos_159 :
    ∃ (c : ℝ) (_ : 0 < c) (C : ℝ),
      ∀ (n : ℕ), 1 ≤ n →
        (SimpleGraph.graphRamsey (SimpleGraph.cycleGraph 4)
          (SimpleGraph.completeGraph (Fin n)) : ℝ) ≤ C * (n : ℝ) ^ (2 - c)

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/159. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/159
[Er78] Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Proc. Ninth Southeastern Conf. Combinatorics, Graph Theory and Computing (1978), 29-40.
[Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.

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.