Duplicate Clique-Sequence Submodularity Conjecture

Definitions and mathematical statement

Let $K_n$ denote the complete simple graph on $n$ vertices, and let $E(K_n)$ be its edge set. The source refers to a "longest $K_t$-sequence rank function" $r_t$ on subsets of $E(K_n)$, but supplies neither the definition of such a sequence nor the rule assigning $r_t(A)$ to an edge set $A$. It also does not specify the admissible integer parameters $n,t$.

Subject to recovering that definition and the intended parameter range, the extracted question is whether, for every admissible $n,t$, \[ r_t(A)+r_t(B)\ge r_t(A\cup B)+r_t(A\cap B) \qquad\text{for all }A,B\subseteq E(K_n). \] This is the submodularity condition. The source further asks whether $r_t$ is the rank function of a matroid on $E(K_n)$. For that conclusion, the intended function must also satisfy \[ r_t(A)\in{\mathbb{Z}},\qquad 0\le r_t(A)\le |A|, \qquad A\subseteq B\Longrightarrow r_t(A)\le r_t(B). \] The supplied record does not establish these additional rank axioms; its use of the term "rank function" cannot replace the missing definition.

The registry marks this extraction as unrecoverable and identifies it as a duplicate of the submodularity clause represented more completely by source record OWR-16633-021. That record's contents are not supplied here. Consequently, this is a conditional statement of the extracted question, with its defining construction and parameter range unresolved.

Short English statement

The extracted question asks whether the longest $K_t$-sequence function is submodular on edge subsets of $K_n$, and hence, subject to the other rank axioms, defines a matroid. The sequence definition and parameter range are missing. The registry retains this record as an unrecoverable duplicate of OWR-16633-021.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30004004$, exact source problem code catalogue entry, title "Duplicate Clique-Sequence Submodularity Conjecture." The supplied verification note identifies OWR-16633-021 as the canonical, more complete representation and states that no separate clean statement is promoted for this duplicate.
• Supplied report citation: Combinatorial Optimization (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/50, pp. 2969--3023. https://doi.org/10.4171/owr/2018/50. The record gives proposed year 2018 and supplies no individual proposer attribution.
• Supplied literature source: Oberwolfach Report 50/2018. https://ems.press/content/serial-article-files/46772.
• Supplied literature source: Abstract 3-Rigidity and Bivariate C2-1-Splines I. https://arxiv.org/abs/1911.00205.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved, reports the extracted conjecture as known for $t=3$ and $t=4$, mentions later rigidity-theoretic partial results, and says that no proof or counterexample for the general statement, including $t=5$, was found. This is supplied dated curation, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.