Diameter Bounds for Connected Layer Families

Definitions and mathematical statement

Let $A$ be a finite alphabet with $|A|=n$. A $d$-element multiset on $A$ is a function $m:A\to{\mathbb{Z}}_{\geq 0}$ satisfying \[ \sum_{a\in A}m(a)=d. \] Thus repetitions of symbols are allowed. The supplied record does not specify the admissible parameter range, define a connected layer family, or define the extremal quantity $\widetilde f(d,n)$. In particular, it does not specify the connectivity condition, the requirements on the layers, or the quantity being maximized. These definitions must be recovered from the cited source before the following question becomes a fully specified mathematical statement.

With those source-dependent definitions and their admissible parameter range retained, the proposed equality is \[ \widetilde f(d,n)=d(n-1). \] The record asks whether this equality holds for every admissible pair $(d,n)$, and specifically whether one can prove or refute \[ \widetilde f(3,n)=3(n-1) \] for every admissible $n$.

A further question concerns layer families arising from triangulated surfaces: are the stated upper bounds $2n$ for surfaces with boundary and $n$ for surfaces without boundary tight? The supplied record does not define the construction associating a layer family to a triangulated surface, identify the bounded quantity, or explicitly identify $n$ in this setting. It also leaves unspecified the allowed surfaces and triangulations and whether tightness means exact attainment for each admissible $n$, attainment for infinitely many $n$, or asymptotic sharpness. Consequently, this part cannot yet be expressed as a precise extremal assertion.

Short English statement

The source asks whether connected layer families of $d$-element multisets on $n$ symbols satisfy $\widetilde f(d,n)=d(n-1)$, especially for $d=3$, and whether the stated surface bounds $2n$ with boundary and $n$ without boundary are sharp. Essential definitions and the meaning of sharpness are missing from the supplied record.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002055$, title "Diameter Bounds for Connected Layer Families," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2012 and names no proposer.
• Supplied report citation: Triangulations (2013), Oberwolfach Reports, pp. 1405--1486, DOI 10.4171/owr/2012/24. https://doi.org/10.4171/owr/2012/24. This is also the supplied literature reference labelled "original OWR report." No problem-specific page or individual author is supplied.
• Supplied literature reference labelled "current diameter paper": https://doi.org/10.1007/s00454-026-00844-8. No author, title, or theorem locator is supplied.
• Supplied literature reference labelled "flag-complex theorem": https://doi.org/10.1287/moor.2014.0661. No author, title, or theorem locator is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the polynomial Hirsch conjecture remains open, that the Hirsch bound is proved for normal flag complexes and related flag cases, and that the general superstrongly-connected family is not settled. This is supplied dated curation, not an independently verified assessment; the record explicitly requests independent verification of time-sensitive status and cited claims. It does not identify which of the precise questions above those results resolve.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that its cleanup is self-contained and adds no unsupported source-dependent content. This supplied editorial qualification does not supply the missing definitions or constitute mathematical verification of this draft. No external source was visited in preparing it.