[#P3018] Monotonicity of the minimum degree forcing a 4-cycle

Problem. Erdős Problem 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$.

A finite mathematical diagram showing a finite graph with a four-cycle marked. — A finite graph with a four-cycle marked.

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Definitions and notation

1Context

This problem originates from Erdős Problem 85 on erdosproblems.com. It concerns the threshold function for the appearance of 4-cycles in graphs with prescribed minimum degree.

2Definitions

Definition 1 (The minimum degree of a graph). The minimum degree of a graph is the smallest number of neighbors among all vertices of the graph.

Definition 2 (A cycle of length $4$, denoted $C_4$). A cycle of length $4$, denoted $C_4$, is a simple graph with four vertices connected in a cyclic sequence.

Definition 3 (The notation $\forall^\infty n$). The notation $\forall^\infty n$ means 'for all sufficiently large $n$', i.e., there exists some $N$ such that the property holds for all $n \geq N$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$.

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 85 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$.^[1]

1Packet records

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 85 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$.	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 85 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$.^[1] Relevance to this problem For erdos problem 85, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 85 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 85: 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 85 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 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$. Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$.	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 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$.^[1] Relevance to this problem For Monotonicity of the minimum degree forcing a 4-cycle, record erdos-problem-85-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 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. 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 85: For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$. 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 85 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 85 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.

ProblemMonotonicity of the minimum degree forcing a 4-cycle

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

“Monotonicity of the minimum degree forcing a 4-cycle.” TheoremDB. P3018. Problem statement; statement text SHA-256 b422240b9deb4f57fd1015a4bc4ad53a3b26b9bd890bb2379a4629deac77a1d9. https://theoremdb.org/statement/?ref=P3018

BibTeX

@misc{theoremdb-problem-b422240b9deb4f57fd1015a4bc4ad53a3b26b9bd890bb2379a4629deac77a1d9,
  title = {{Monotonicity of the minimum degree forcing a 4-cycle}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 b422240b9deb4f57fd1015a4bc4ad53a3b26b9bd890bb2379a4629deac77a1d9},
  url = {https://theoremdb.org/statement/?ref=P3018}
}

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.

1Lean verification

1References

Packet source. Erdős Problems database, Problem 85, maintained status record. Erdős Problems record 85, checked 2026-08-01. Problem 85; 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 85; 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 Monotonicity of the minimum degree forcing a 4-cycle: 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 85 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 Monotonicity of the minimum degree forcing a 4-cycle: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 85. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/85.lean:L40; theorem erdos_85; 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 Monotonicity of the minimum degree forcing a 4-cycle: 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.

[#P3018] Monotonicity of the minimum degree forcing a 4-cycle

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