Extremal Conjecture for Partially Directed Uniform Hypergraphs

Definitions and mathematical statement

Let $k\geq 2$ be an integer. The supplied record uses the terms $k$-PDG, $\widetilde T_k$, and the quantities $\alpha$ and $\beta$, but does not define them. In particular, it does not specify the permitted partially directed hyperedges, edge multiplicities, the normalization of $\alpha$ and $\beta$, the forbidden configuration $\widetilde T_k$, or the precise meaning of subgraph containment for these objects. These definitions must be recovered from the cited source before the conjecture has a self-contained mathematical formulation.

Subject to those unresolved source definitions, the target is the following statement, identified in the report excerpt as Conjecture 4: for every integer $k\geq 2$, there exists an integer $n_0(k)\geq 1$ such that, for every integer $n\geq n_0(k)$, every $n$-vertex $k$-PDG containing no subgraph isomorphic to $\widetilde T_k$ satisfies \[ \alpha+(\log_2 3)\beta\leq 1. \] Here $\alpha$ and $\beta$ are the unspecified quantities associated with that $k$-PDG in the source. The dependence of the threshold on $k$ and the exact inequality are retained; no asymptotic error term is introduced.

The excerpt additionally reports, as Theorem 5, that for each fixed $k\geq 2$, Conjecture 4 implies its Conjecture 1. It also reports an essential equivalence with Conjecture 1 after replacing the single forbidden configuration by a special finite family $F_k$. Neither Conjecture 1, the family $F_k$, nor the meaning of that essential equivalence is supplied, so these contextual claims cannot be formulated precisely here.

Short English statement

For each fixed integer $k\geq 2$, the conjecture asks whether every sufficiently large $k$-PDG avoiding $\widetilde T_k$ satisfies $\alpha+(\log_2 3)\beta\leq 1$. The supplied record omits the definitions of the objects, forbidden configuration, and quantities in this inequality, so this is a conditional transcription requiring source clarification and human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005110$, source problem code catalogue entry, titled "Extremal Conjecture for Partially Directed Uniform Hypergraphs," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2022 and no named proposer.
• Supplied report citation: Combinatorics, Probability and Computing (2023), Oberwolfach Reports, pp. 1165--1237, DOI 10.4171/owr/2022/22. https://doi.org/10.4171/owr/2022/22. The supplied excerpt identifies the target as Conjecture 4 and its implication to Conjecture 1 as Theorem 5. Oberwolfach Reports are described in the record as documenting mathematical workshops and their open-problem discussions. The supplied citation year 2023 and proposed year 2022 are retained without reconciliation.
• Supplied literature reference: Dong, Mani, and Zhao, Enumerating k-SAT functions. https://arxiv.org/abs/2107.09233. The report excerpt attributes its $k=4$ proof to an application of a result of Füredi and Melaki, labeled [5] there, on minimum triangular edge density at fixed graph edge density. It attributes the $k=5$ brute-force verification to an appendix by Nitya Mani and Edward Yu in arXiv-v2 of its reference [4]. The excerpt directs readers to [4] for the precise finite-forbidden-family formulation and its construction. Complete bibliography entries for those numbered references are not supplied; their identification is not assumed here.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, reports the conjecture established for $k=4$ and computationally checked for $k=5$, and states that no proof for every fixed $k$ is known. This is supplied dated curation, not an independently verified literature assessment. The record explicitly asks that time-sensitive status and cited claims be verified independently.
• The record labels its statement corrected_verified and says that the official report was checked directly and that the clean statement restores definitions and removes extraction artifacts. These are supplied verification claims; no source visit or independent verification is asserted in this draft. The supplied text itself does not include the necessary definitions.