[#P3068] Minimum avoiding alphabet for every avoidable word

Problem. Let \(u\) be a finite word using \(c(u)\) distinct variables. Define \(m(u)\) as the least alphabet size admitting an infinite word with no contiguous factor equal to \(\phi(u)\) for any nonerasing morphism \(\phi\). Determine \(m(u)\) for every avoidable word \(u\).

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Definitions and notation

1Context

Known frontier: General estimates and values for special pattern classes are known. Open boundary: Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u.

2Problem setup

Definition 1 (avoidable word). A pattern u is avoidable when such an infinite word exists over some finite alphabet.

Definition 2 (nonerasing morphism). A morphism from pattern variables to nonempty finite words, extended to a pattern by concatenation.

Remark 1. This is Problem 2.5 in the 2026 Lyapin Notebook; the public formulation fixes the input and equivalence conventions needed for independent review.

3What counts as a solution

Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: General estimates and values for special pattern classes are known. Exact unresolved remainder: Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u.^[1][2]

1Packet records

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: General estimates and values for special pattern classes are known. Exact unresolved remainder: Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: General estimates and values for special pattern classes are known. Exact unresolved remainder: Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Minimum avoiding alphabet for every avoidable word. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: General estimates and values for special pattern classes are known. The exact unresolved remainder is: Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u. A complete resolution must meet the following acceptance conditions: - Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u. Open full record →
General estimates and values for special pattern classes are known.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 General estimates and values for special pattern classes are known.^[1][2] Relevance to this problem Locates the present research frontier immediately below Minimum avoiding alphabet for every avoidable word. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details General estimates and values for special pattern classes are known. This leaves the following boundary unresolved: Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.^[1][2] Relevance to this problem Documents why Minimum avoiding alphabet for every avoidable word was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across source collection and its cited primary literature, Crossref/arXiv or MathSciNet stable records, TheoremDB published, prospecting, formal-conjecture, packet, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - "Lyapin notebook" "Problem 2.5" - "Minimum avoiding alphabet for every avoidable word" - site:arxiv.org semigroup "avoidable word" open problem The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there. Related: Claim 1 Open full record →
Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u. Relevance to this problem Turns the remaining uncertainty in Minimum avoiding alphabet for every avoidable word into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: General estimates and values for special pattern classes are known. Exact unresolved remainder: Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u.

Equivalent-formulation queries: "Lyapin notebook" "Problem 2.5"; "Minimum avoiding alphabet for every avoidable word"; site:arxiv.org semigroup "avoidable word" open problem
Strongest checked neighboring result: General estimates and values for special pattern classes are known.
Exact unresolved remainder: Give a formula or effective complete classification returning the exact integer m(u) for every avoidable u.

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ProblemMinimum avoiding alphabet for every avoidable word

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2See also

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Cycle Double Cover Conjecturecombinatorics

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“Minimum avoiding alphabet for every avoidable word.” TheoremDB. P3068. Problem statement; statement text SHA-256 4e7bc704220e4dac554b99b5d34086d1a35b201e6f267364e547722c3d15b765. https://theoremdb.org/statement/?ref=P3068

BibTeX

@misc{theoremdb-problem-4e7bc704220e4dac554b99b5d34086d1a35b201e6f267364e547722c3d15b765,
  title = {{Minimum avoiding alphabet for every avoidable word}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 4e7bc704220e4dac554b99b5d34086d1a35b201e6f267364e547722c3d15b765},
  url = {https://theoremdb.org/statement/?ref=P3068}
}

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This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. Bershadsky, S., Kublanovsky, S., and Mashevitzky, G., “The Lyapin's notebook: a collection of unsolved problems in Semigroup Theory”. arXiv (2026). DOI 10.48550/arXiv.2604.04763. Problem 2.5 and its immediately following status paragraph. ↗ open copy ↗preprint · primary source · arXiv:2604.04763, checked 2026-08-01 · checked 2026-08-01Source use: original summary.States the numbered open problem, identifies its proposer, and summarizes known special cases.Also cited at S. Bershadsky, S. Kublanovsky, and G. Mashevitzky, “The Lyapin’s notebook: a collection of unsolved problems in Semigroup Theory,” arXiv:2604.04763 (2026). Problem 2.5 and its immediately following status paragraph.Source used to assess the problem's recorded status.For Minimum avoiding alphabet for every avoidable word: This is the dated publication status for the canonical target Minimum avoiding alphabet for every avoidable word.Source named by the research packet.
Kirby A. Baker, George F. McNulty, and Walter Taylor, “Growth problems for avoidable words”. Theoretical Computer Science 69(3) (1989), 319-345. DOI 10.1016/0304-3975(89)90071-6. Bounds for avoidable patterns. ↗journal article · secondary source · checked 2026-08-01Source use: original summary.Provides established estimates and foundational definitions for the exact alphabet-size problem.Source used to assess the problem's recorded status.For Minimum avoiding alphabet for every avoidable word: Provides established estimates and foundational definitions for the exact alphabet-size problem.

Original TheoremDB statement and summary based on citation-only scholarly sources; no source prose, proof, table, code, or figure is reproduced.

[#P3068] Minimum avoiding alphabet for every avoidable word

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