Extremal Subgraph Counts in Sparse Graph Classes

Definitions and mathematical statement

For a graph $H$ and a graph $G$, let \[ N_H(G)=\#\{F:F\text{ is a subgraph of }G\text{ and }F\cong H\}. \] This follows the supplied clean statement's wording of subgraphs isomorphic to $H$; the counting convention and permitted graph types require confirmation from the source. For a graph class $\mathcal G$ and a nonnegative integer $n$, write \[ \mathcal G_n=\{G\in\mathcal G:|V(G)|=n\}. \] Whenever $\mathcal G_n$ is nonempty and the indicated maximum exists, the requested extremal quantity is \[ \operatorname{ex}(H,\mathcal G,n) =\max_{G\in\mathcal G_n}N_H(G). \] Requested determination. Determine $\operatorname{ex}(H,\mathcal G,n)$ for the intended graph $H$, graph class $\mathcal G$, and admissible vertex counts $n$.

Unresolved qualifications. The supplied record specifies neither $H$ nor $\mathcal G$, and supplies no definition of the robust sparsity mentioned in its contextual talk title. It also leaves graph types, counting conventions, the range of $n$, and the treatment of $\mathcal G_n=\varnothing$ unspecified. Consequently this is a conditional description of the requested extremal quantity, not a fully specified extremal problem. No value is assigned when the defining maximum is unavailable.

Short English statement

The question asks for the largest number of copies of a graph $H$ that can occur in an $n$-vertex graph belonging to a class $\mathcal G$. The supplied record does not identify $H$ or $\mathcal G$, so the intended concrete problem remains unresolved. Its open status is a supplied curation label.

Sources

• Local UnsolvedMath registry, permanent ID $30004966$, title Extremal Subgraph Counts in Sparse Graph Classes, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems.
• Supplied source citation: Graph Theory (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/1, pp. 5--78. Supplied source URL: https://doi.org/10.4171/owr/2022/1. The supplied literature reference labels this same link Oberwolfach Report 1/2022.
• The original extracted statement appends the contextual title Homomorphism counts in robustly sparse graphs and the name Chun-Hung Liu. The record does not explicitly identify a problem proposer; its proposed-year field is $2022$.
• Literature assessment dated 2026-08-22: supplied dated curation, not independently verified here. It labels the problem open because the extracted record does not define $\mathcal G$ or $H$, preventing verification of a determinate extremal claim. Its listed source is Oberwolfach Report 1/2022, https://doi.org/10.4171/owr/2022/1. The background requests independent verification of time-sensitive status and cited claims.
• The record labels the statement corrected_verified and reports verification against the official Oberwolfach report. This is supplied verification metadata, not an independent verification performed for this draft.