[#P3016] Asymptotic growth of the maximum guaranteed order of a regular induced subgraph

Problem. Erdős Problem 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here, a regular induced subgraph of a graph \(G\) is an induced subgraph that is \(k\)-regular for some non-negative integer \(k\), meaning every vertex in the subgraph has exactly \(k\) neighbors within the subgraph. Determine whether \(\displaystyle \frac{F(n)}{\log n} \to \infty\) as \(n \to \infty\).

A finite mathematical diagram showing an induced regular subgraph marked inside a graph. — An induced regular subgraph marked inside a finite graph.

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1Context

This problem concerns the extremal function for regular induced subgraphs in graph theory, a topic initiated by Erdős. The function \(F(n)\) measures the worst-case guarantee for finding large regular induced subgraphs. The best known upper bound, due to Alon, Krivelevich, and Sudakov (2007), establishes that \(F(n) = O\bigl(n^{1/2} (\log n)^{3/4}\bigr)\). The question of whether \(F(n)\) grows faster than any constant multiple of \(\log n\) remains open.

2Problem setup

Definition 1 (A simple graph on a vertex set \(V\). A simple graph on a vertex set \(V\) is an undirected graph without loops or multiple edges.

Definition 2 (An induced subgraph of a graph \(G = (V, E)\) on a subset \(S \subseteq V\). An induced subgraph of a graph \(G = (V, E)\) on a subset \(S \subseteq V\) is the graph with vertex set \(S\) whose edges are exactly those edges of \(G\) with both endpoints in \(S\).

Definition 3 (A graph). A graph is \(k\)-regular if every vertex has degree exactly \(k\).

Remark 1. For functions \(f, g : \mathbb{N} \to \mathbb{R}_{>0}\), we write \(f(n) = O(g(n))\) if there exist constants \(C > 0\) and \(n_0 \in \mathbb{N}\) such that \(f(n) \leq C \cdot g(n)\) for all \(n \geq n_0\).

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here, a regular induced subgraph of a graph \(G\) is an induced subgraph that is \(k\)-regular for some non-negative integer \(k\), meaning every vertex in the subgraph has exactly \(k\) neighbors within the subgraph. Determine whether \(\displaystyle \frac{F(n)}{\log n} \to \infty\) as \(n \to \infty\).

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 82 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here, a regular induced subgraph of a graph \(G\) is an induced subgraph that is \(k\)-regular for some non-negative integer \(k\), meaning every vertex in the subgraph has exactly \(k\) neighbors within the subgraph. Determine whether \(\displaystyle \frac{F(n)}{\log n} \to \infty\) as \(n \to \infty\).^[1]

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 82 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here, a regular induced subgraph of a graph \(G\) is an induced subgraph that is \(k\)-regular for some non-negative integer \(k\), meaning every vertex in the subgraph has exactly \(k\) neighbors within the subgraph. Determine whether \(\displaystyle \frac{F(n)}{\log n} \to \infty\) as \(n \to \infty\).	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 82 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here, a regular induced subgraph of a graph \(G\) is an induced subgraph that is \(k\)-regular for some non-negative integer \(k\), meaning every vertex in the subgraph has exactly \(k\) neighbors within the subgraph. Determine whether \(\displaystyle \frac{F(n)}{\log n} \to \infty\) as \(n \to \infty\).^[1] Relevance to this problem For erdos problem 82, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 82 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 82: 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 82 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 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here, a regular induced subgraph of a graph \(G\) is an induced subgraph that is \(k\)-regular for some non-negative integer \(k\), meaning every vertex in the subgraph has exactly \(k\) neighbors within the subgraph. Determine whether \(\displaystyle \frac{F(n)}{\log n} \to \infty\) as \(n \to \infty\). Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here, a regular induced subgraph of a graph \(G\) is an induced subgraph that is \(k\)-regular for some non-negative integer \(k\), meaning every vertex in the subgraph has exactly \(k\) neighbors within the subgraph. Determine whether \(\displaystyle \frac{F(n)}{\log n} \to \infty\) as \(n \to \infty\).	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 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here, a regular induced subgraph of a graph \(G\) is an induced subgraph that is \(k\)-regular for some non-negative integer \(k\), meaning every vertex in the subgraph has exactly \(k\) neighbors within the subgraph. Determine whether \(\displaystyle \frac{F(n)}{\log n} \to \infty\) as \(n \to \infty\).^[1] Relevance to this problem For Asymptotic growth of the maximum guaranteed order of a regular induced subgraph, record erdos-problem-82-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 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. 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 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here, a regular induced subgraph of a graph \(G\) is an induced subgraph that is \(k\)-regular for some non-negative integer \(k\), meaning every vertex in the subgraph has exactly \(k\) neighbors within the subgraph. Determine whether \(\displaystyle \frac{F(n)}{\log n} \to \infty\) as \(n \to \infty\). 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 82 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 82 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.

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ProblemAsymptotic growth of the maximum guaranteed order of a regular induced subgraph

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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“Asymptotic growth of the maximum guaranteed order of a regular induced subgraph.” TheoremDB. P3016. Problem statement; statement text SHA-256 b532bc1cbeb0e031f508bef75c6f1f8d31026b3f51349435a6c95bd81dc47262. https://theoremdb.org/statement/?ref=P3016

BibTeX

@misc{theoremdb-problem-b532bc1cbeb0e031f508bef75c6f1f8d31026b3f51349435a6c95bd81dc47262,
  title = {{Asymptotic growth of the maximum guaranteed order of a regular induced subgraph}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 b532bc1cbeb0e031f508bef75c6f1f8d31026b3f51349435a6c95bd81dc47262},
  url = {https://theoremdb.org/statement/?ref=P3016}
}

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

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1References

Packet source. Erdős Problems database, Problem 82, maintained status record. Erdős Problems record 82, checked 2026-08-01. Problem 82; 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 82; 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 Asymptotic growth of the maximum guaranteed order of a regular induced subgraph: 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 82 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 Asymptotic growth of the maximum guaranteed order of a regular induced subgraph: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 82. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/82.lean:L49; theorem erdos_82; 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 Asymptotic growth of the maximum guaranteed order of a regular induced subgraph: 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.

[#P3016] Asymptotic growth of the maximum guaranteed order of a regular induced subgraph

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