Erdős (Erdos) Problem #805: status in 2026

Erdős Problem #805

For which functions g(n) with n>g(n)≥ (log n)^2 is there a graph on n vertices in which every induced subgraph on g(n) vertices contains a clique of size ≥ log n and an independent set of size ≥ log n? In particular, is there such a graph for g(n)=(log n)^3? This is the first part.

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 #805 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #805 on erdosproblems.com	open 2025-08-31	erdosproblems.com
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.

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Cite

@misc{cairn-erdos-805,
  title        = {Erdős Problem #805},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-805}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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

erdos_805.parts.i. For which functions $g (n)$ with $n > g (n) \geq (lo g n)^{2}$ is there a graph on $n$ vertices in which every induced subgraph on $g (n)$ vertices contains a clique of size $\geq lo g n$ and an independent set of size $\geq lo g n$ ?

In particular, is there such a graph for $g (n) = (lo g n)^{3}$ ?

This is the first part. A problem of Erdős and Hajnal [Er91]; see also erdosproblems.com/804. The inequalities on g are required for all large n, and a graph is required for all large n.

erdos_805.parts.ii. For which functions $g (n)$ with $n > g (n) \geq (lo g n)^{2}$ is there a graph on $n$ vertices in which every induced subgraph on $g (n)$ vertices contains a clique of size $\geq lo g n$ and an independent set of size $\geq lo g n$ ?

In particular, is there such a graph for $g (n) = (lo g n)^{3}$ ?

This is the second part. Erdős and Hajnal thought that there is no such graph for $g (n) = (lo g n)^{3}$ [AlSu07]; we state the positive proposition, with $g (n) = ⌊(lo g n)^{3} ⌋$ and a graph required for all large $n$ , as in the first part.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«805» (2 statements).

theorem erdos_805.parts.i :
    {g : ℕ → ℕ | (∀ᶠ n : ℕ in atTop, Real.log n ^ 2 ≤ g n ∧ g n < n) ∧ HasRamseyGraphs g} =
      erdos_805.parts.i.answer

theorem erdos_805.parts.ii : HasRamseyGraphs (fun n ↦ ⌊Real.log n ^ 3⌋₊)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/805. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/805
[Er91] Erdős, P., *Problems and results in combinatorial analysis and combinatorial number theory*. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.
[AlSu07] Alon, N. and Sudakov, B., *On graphs with subgraphs having large independence numbers*. J. Graph Theory 56 (2007), 149-157. arXiv:0706.4099
[ABS21] Alon, N., Bucić, M. and Sudakov, B., *Large cliques and independent sets all over the place*. Proc. Amer. Math. Soc. 149 (2021), 3145-3157. arXiv:2004.04718

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.