Extremal Geometric Graphs with Forbidden Configurations

Definitions and mathematical statement

Let $H$ be a family of forbidden geometric configurations, and let $n$ be an admissible integer vertex count. The source does not specify the geometric graph model or the meaning of containment of a configuration. Accordingly, the following notation is conditional on fixing those conventions.

Write $\mathcal G_n$ for the class of geometric graphs with exactly $n$ vertices in the chosen model, and write $F\preccurlyeq G$ when the configuration $F$ occurs in $G$ according to the chosen containment relation. A graph $G$ is $H$-free if \[ \forall F\in H,\qquad F\not\preccurlyeq G. \] Define the set of attainable edge counts by \[ A_H(n)=\left\{|E(G)|:G\in\mathcal G_n,\ \forall F\in H,\ F\not\preccurlyeq G\right\}. \] Problem. Given $H$, determine the maximum number of edges in an $H$-free geometric graph with $n$ vertices; in the notation above, determine \[ \operatorname{ex}_{\mathcal G,\preccurlyeq}(n,H)=\max A_H(n) \] whenever this maximum exists.

This is a conditional formulation of the supplied question. No particular family $H$, ambient space, drawing convention, general-position hypothesis, or containment relation is supplied. The source also gives no convention for an empty admissible class or for a failure of a finite maximum to exist. These choices must be resolved before the question defines a particular extremal function.

Short English statement

For a given forbidden family $H$, determine how many edges an $n$-vertex geometric graph can have while avoiding every configuration in $H$. The geometric model and the meaning of occurrence must first be specified; the supplied record leaves them unresolved.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001890$, title "Extremal Geometric Graphs with Forbidden Configurations," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2011.
• Supplied source citation: Discrete Geometry (2012), Oberwolfach Reports, pp. 2459--2548, DOI 10.4171/owr/2011/44. Supplied report link and literature-source link, labeled "OWR 44 (2012)": https://doi.org/10.4171/owr/2011/44. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; it supplies no exact problem page or individual attribution.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the source frames the extremal geometric-graph problem for general forbidden configurations, with no general asymptotic formula matching the extracted question located. This is supplied dated literature triage, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, but its supplied verification explains only that math delimiters were added around the forbidden family and vertex-count parameter without changing the question. This does not establish mathematical completeness or independent source verification.