Erdős (Erdos) Problem #563: status in 2026

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.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #563 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_563 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #563 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_563`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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.

Try it with your chatbot Submit a claim Follow (RSS)

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 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Ferdos-563.json)](https://cairn-commons.com/problems/erdos-563)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.