Erdős Problem #566 — open problem in graph theory

Erdős Problem #566

Let G be such that any subgraph on k vertices has at most 2k-3 edges. Is it true that, if H has m edges and no isolated vertices, then R(G,H) ≪ m? In other words: if G is sparse (every induced subgraph on k vertices has ≤ 2k-3 edges), is G Ramsey size linear?

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.

Start working on it Submit a claim Follow

Cite

@misc{cairn-erdos-566,
  title        = {Erdős Problem #566},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-566}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $G$ be such that any subgraph on $k$ vertices has at most $2 k - 3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $R (G, H) ≪ m$ ?

In other words: if $G$ is sparse (every induced subgraph on $k$ vertices has $\leq 2 k - 3$ edges), is $G$ Ramsey size linear?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«566». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_566 : answer(sorry) ↔
    ∀ (p : ℕ) (G : SimpleGraph (Fin p)),
      (∀ S : Finset (Fin p), 2 ≤ S.card → (G.induce S).edgeSet.ncard ≤ 2 * S.card - 3) →
      G.IsRamseySizeLinear

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/566. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/566
[EFRS93] Erdős, Faudree, Rousseau and Schelp, _Ramsey size linear graphs_.

Combin. Probab. Comput. (1993), 389-399.

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.