Erdős (Erdos) Problem #560: status in 2026

Erdős Problem #560

Let hatR(G) denote the size Ramsey number, the minimal number of edges m such that there is a graph H with m edges such that in any 2-colouring of the edges of H there is a monochromatic copy of G. Determine hatR(K_n,n), where K_n,n is the complete bipartite graph with n vertices in each component.

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 #560 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_560 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #560 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_560`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 2 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-560,
  title        = {Erdős Problem #560},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-560}},
  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-560.json)](https://cairn-commons.com/problems/erdos-560)

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 $R^(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$ -colouring of the edges of $H$ there is a monochromatic copy of $G$ .

Determine $R^(K_{n, n}),$ where $K_{n, n}$ is the complete bipartite graph with $n$ vertices in each component.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«560».

theorem erdos_560 : erdos_560.answer = (fun n : ℕ ↦ r̂(n))

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/560. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_560.variants.conlon_fox_wigderson — Conlon, Fox, and Wigderson [CFW23] conjecture that hatR(K_s,t)asymp s^2t2^s for all s≤ t, and so in particular we should have hatR(K_n,n)asymp n^32^n.

References

erdosproblems.com/560
[EFRS82] Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey numbers for brooms. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (1982), 283-293.
[ErRo93] Erdős, P. and Rousseau, C. C., The size Ramsey number of a complete bipartite graph. Discrete Math. (1993), 259-262.
[EFRS78b] Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., The size Ramsey number. Period. Math. Hungar. (1978), 145-161.
[NeRo78] Nešetřil, J. and Rödl, V., The structure of critical Ramsey graphs. Acta Math. Acad. Sci. Hungar. (1978), 295-300.
[CFW23] Conlon, D. and Fox, J. and Wigderson, Y., Three early problems on size Ramsey numbers. Combinatorica (2023), 743-768.

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.