Erdős Problem #563 — open problem in combinatorics

Erdős Problem #563

Let F(n,α) denote the smallest m such that there exists a 2-colouring of the edges of K_n so that every X⊆ [n] with lvert Xrvert≥ m contains more than α C(lvert Xrvert, 2) many edges of each colour. Prove that, for every 0≤ α < 1/2, F(n,α)∼ c_αlog n for some constant c_α depending only on α.

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-563,
  title        = {Erdős Problem #563},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-563}},
  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

Let $F (n, α)$ denote the smallest $m$ such that there exists a $2$ -colouring of the edges of $K_{n}$ so that every $X \subseteq [n]$ with $∣ X ∣ \geq m$ contains more than $α (2 ∣ X ∣)$ many edges of each colour.

Prove that, for every $0 \leq α < 1/2$ , $F (n, α) \sim c_{α} lo g n$ for some constant $c_{α}$ depending only on $α$ .

This problem is #39 in Ramsey Theory in the graphs problem collection.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«563».

theorem erdos_563 :
    ∀ (α : ℝ), 0 ≤ α → α < 1 / 2 →
      ∃ (c : ℝ), 0 < c ∧
        Tendsto (fun n : ℕ => (F n α : ℝ) / Real.log n) atTop (nhds 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/563. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/563
[Er90b] Erdős, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28.

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.