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

- ID: `P3018`
- Reference: `erdos-problem-85`
- Page: https://theoremdb.org/statements/P3018
- Record maturity: Reviewed problem with recorded work

## 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 $n$.

### Context

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.

### Problem setup

- **Definition (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 (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 (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 $n \geq N$.
- **Remark.** 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.

### What 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 $n$.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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 $n$.

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 $n$.

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-85`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 85, maintained status record. Erdős Problems record 85, checked 2026-08-01. Problem 85; status field and linked bibliography https://www.erdosproblems.com/85
   - 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 85 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 85. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/85.lean:L40; theorem erdos_85; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/85.lean#L40
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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.