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Level A · Machine-checkable Hard Graph theory P-erdos-545

Erdős Problem #545

Let m be sufficiently large and let G be a graph with m edges and no isolated vertices. Is the Ramsey number R(G) maximised when G is 'as complete as possible'?

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-545,
  title        = {Erdős Problem #545},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-545}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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Verified
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Disputed
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Refuted
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On the literature board
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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

Let be sufficiently large and let be a graph with edges and no isolated vertices. Is the Ramsey number maximised when is 'as complete as possible'? That is, if edges with then is where is the graph formed by connecting a new vertex to of the vertices of ?

A question of Erdős and Graham. The restriction to sufficiently large excludes the small counterexamples recorded on the source page.

This problem is #10 in Ramsey Theory in the graphs problem collection.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«545». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_545 : answer(sorry) ↔
    ∀ᶠ m : ℕ in atTop, ∀ (n t : ℕ), t < n → m = n.choose 2 + t →
      ∀ (V : Type) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj],
        (∀ v, 0 < G.degree v) →
        G.edgeSet.ncard = m →
        SimpleGraph.diagonalGraphRamsey G ≤
          SimpleGraph.diagonalGraphRamsey (knPlusTEdges n t)

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/545. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/545
  • [ErGr75] Erdős, P. and Graham, R. L., On partition theorems for finite graphs. Infinite and finite sets (1975), 515-527.
  • [Er84b] Erdős, P., On some problems in graph theory, combinatorial analysis and combinatorial number theory. Graph theory and combinatorics (Cambridge, 1983) (1984), 1-17.

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.