[#P3004] The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds

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Problem. Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \mathbb{N} \to \mathbb{R}$, say that $f$ is an \textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\alpha(G) \geq f(n)$ or $\omega(G) \geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$?

A finite mathematical diagram showing a graph with a clique and an independent set marked. — A finite graph with two homogeneous vertex sets marked.

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Definitions and notation

1Context

This problem asks whether the Erdős–Hajnal conjecture holds: that forbidding any fixed induced subgraph forces a polynomially large homogeneous set (clique or independent set), rather than merely the logarithmic bound guaranteed by Ramsey's theorem.

2Definitions

Definition 1 (A simple graph $G$). A simple graph $G$ consists of a vertex set together with a set of undirected edges, with no loops and no multiple edges.

Definition 2 (A graph $G$). A graph $G$ is $H$-free if there is no embedding of the vertices of $H$ into the vertices of $G$ that preserves both adjacency and non-adjacency (i.e., $H$ is not an induced subgraph of $G$).

Definition 3 (The independence number $\alpha(G)$ of a graph $G$). The independence number $\alpha(G)$ of a graph $G$ is the maximum cardinality of a set of vertices with no edges between them.

Definition 4 (The clique number $\omega(G)$ of a graph $G$). The clique number $\omega(G)$ of a graph $G$ is the maximum cardinality of a set of vertices in which every pair is joined by an edge.

Definition 5 (An Erdős–Hajnal lower bound for $H$). An Erdős–Hajnal lower bound for $H$ is a function $f$ such that for all sufficiently large $n$, every $n$-vertex $H$-free graph has either independence number or clique number at least $f(n)$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \mathbb{N} \to \mathbb{R}$, say that $f$ is an \textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\alpha(G) \geq f(n)$ or $\omega(G) \geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$?

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 61 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \mathbb{N} \to \mathbb{R}$, say that $f$ is an \textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\alpha(G) \geq f(n)$ or $\omega(G) \geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$?^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 61 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \mathbb{N} \to \mathbb{R}$, say that $f$ is an \textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\alpha(G) \geq f(n)$ or $\omega(G) \geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$?	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 61 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \mathbb{N} \to \mathbb{R}$, say that $f$ is an \textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\alpha(G) \geq f(n)$ or $\omega(G) \geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$?^[1] Relevance to this problem For erdos problem 61, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 61 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 61 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \mathbb{N} \to \mathbb{R}$, say that $f$ is an \textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\alpha(G) \geq f(n)$ or $\omega(G) \geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \mathbb{N} \to \mathbb{R}$, say that $f$ is an \textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\alpha(G) \geq f(n)$ or $\omega(G) \geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$?	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \mathbb{N} \to \mathbb{R}$, say that $f$ is an \textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\alpha(G) \geq f(n)$ or $\omega(G) \geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$?^[1] Relevance to this problem For The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds, record erdos-problem-61-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \mathbb{N} \to \mathbb{R}$, say that $f$ is an \textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\alpha(G) \geq f(n)$ or $\omega(G) \geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 61 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 61 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemThe Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Constructing a union from non-cancelling intersectionscombinatorics
Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

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Plain text

“The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds.” TheoremDB. P3004. Problem statement; statement text SHA-256 2ac18e10aeece44ca26e85a641e3addb74658ab7db46a3ecdf1854d5647ccf6a. https://theoremdb.org/statement/?ref=P3004

BibTeX

@misc{theoremdb-problem-2ac18e10aeece44ca26e85a641e3addb74658ab7db46a3ecdf1854d5647ccf6a,
  title = {{The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 2ac18e10aeece44ca26e85a641e3addb74658ab7db46a3ecdf1854d5647ccf6a},
  url = {https://theoremdb.org/statement/?ref=P3004}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1References

Packet source. Erdős Problems database, Problem 61, maintained status record. Erdős Problems record 61, checked 2026-08-01. Problem 61; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 61; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 61 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 61. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/61.lean:L46; theorem erdos_61; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

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[#P3004] The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds

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