Maximal Strongly Free Quadratic Relator Sequences

Definitions and mathematical statement

For a positive integer $d$, consider the source's strongly free sequences of relators on $d$ generators, with every relator of degree two. The ambient algebraic or pro-$p$ presentation category, the grading, the coefficient field, and the definition of strongly free are not supplied. Length means the number of relators in the sequence.

The conjectured largest possible length is exactly \[ t(d)= \begin{cases} d^2/4,&d\text{ even},\\[2pt] (d-1)^2/4,&d\text{ odd}. \end{cases} \] Once the intended category is fixed, the maximality assertion has two parts: there exists an admissible strongly free quadratic-relator sequence of length $t(d)$, and every such sequence has length at most $t(d)$. The supplied assessment attributes a construction and the displayed quadratic-degree bound to the report, but says maximality is unproved.

The odd case is reproduced as $(d-1)^2/4$, not changed to $(d^2-1)/4$ or $\lfloor d^2/4\rfloor$. The report must confirm the category and hypotheses supporting this particular parity-dependent expression. In particular, notions of strongly free sequences for associative algebras, Lie or restricted Lie algebras, and pro-$p$ presentations cannot be silently identified. The meaning of quadratic relators also depends on which grading or filtration is used.

The assessment additionally mentions unresolved nil and self-similar algebra questions in remaining characteristics, but gives no characteristic range or precise connection to this maximal-length problem. Those statements are recorded as context rather than used to supply missing hypotheses or formulate another problem.

Short English statement

The question asks whether the displayed parity-dependent value is the exact maximum number of strongly free quadratic relators on $d$ generators. The odd formula is preserved exactly; the ambient category, strong-freeness definition, and characteristic assumptions are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000490, code catalogue entry, "Maximal Strongly Free Quadratic Relator Sequences." Proposed year 2006; proposer unspecified.
• Pro-$p$ Extensions of Global Fields and pro-$p$ Groups (2007), Oberwolfach Reports, pp. 1463--1536, DOI 10.4171/owr/2006/25; https://doi.org/10.4171/owr/2006/25. The displayed report year 2007 differs from the DOI and proposed year 2006; the report title's escaped dollar signs are normalized as mathematical $p$.
• Supplied reference "Petrogradsky--Shestakov open problems": https://jolt.centre-mersenne.org/item/10.5802/jolt.731.pdf.
• Statement status corrected_verified; the supplied verification says editorial and LaTeX cleanup preserved meaning and added no source-dependent content.
• Literature triage checked 2026-08-21: open. It reports the construction and bound $t(d)$ without proved maximality, and mentions related unresolved nil/self-similar algebra questions in unspecified remaining characteristics. Exact relevance and status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.