Piercing Line Families with the (p,3)-Property

Definitions and mathematical statement

Let $X$ denote the intended ambient point space and let $\mathcal L$ denote its collection of straight lines. The supplied record does not specify the ambient geometry or dimension; these remain unresolved parameters, rather than a claim about arbitrary incidence structures.

For an integer $p\geq 3$, a finite family $\mathcal F\subseteq\mathcal L$ has the $(p,3)$-property if \[ \forall\mathcal G\subseteq\mathcal F,\quad |\mathcal G|=p\ \Longrightarrow\ \exists\ell_1,\ell_2,\ell_3\in\mathcal G\text{ pairwise distinct},\quad \ell_1\cap\ell_2\cap\ell_3\ne\varnothing. \] Thus concurrency means that the three lines contain a common point of $X$. A point set $U\subseteq X$ pierces $\mathcal F$ if \[ \forall\ell\in\mathcal F,\qquad \ell\cap U\ne\varnothing. \] The supplied question asks whether, for every integer $p\geq3$ and every such finite family with the $(p,3)$-property, there exists a piercing set satisfying \[ \exists U\subseteq X,\qquad |U|\leq p-2 \quad\text{and}\quad \forall\ell\in\mathcal F,\ \ell\cap U\ne\varnothing. \] Here families are represented as sets of distinct lines; the source does not explicitly discuss repetitions. If $|\mathcal F|<p$, the displayed $(p,3)$-property is vacuous. The record supplies neither a restriction $|\mathcal F|\geq p$ nor a separate convention for these small families, so the intended boundary formulation requires review. No such restriction has been inserted. The finite-family qualification comes from the supplied cleaned statement; the supplied original wording does not explicitly impose finiteness.

Short English statement

For each integer $p\geq3$, must a finite family of straight lines in which every $p$ lines contain three concurrent lines be pierced by at most $p-2$ points? The ambient geometry and the treatment of families with fewer than $p$ lines are unspecified and require clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003460$, source problem code catalogue entry, title "Piercing Line Families with the $(p,3)$-Property." The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied source citation: Discrete Geometry (2018), Oberwolfach Reports, pp. 1135--1205, DOI 10.4171/owr/2017/19. Original Oberwolfach report: https://doi.org/10.4171/owr/2017/19. The background attributes the problem to mathematical workshop open-problem discussions documented in Oberwolfach Reports. The supplied 2018 citation and 2017 year metadata require reconciliation.
• Supplied literature source: "Ramsey type problems for lines and points," https://www.renyi.hu/conferences/hunrus18/abstracts/grigoriev_abstract.pdf. No author attribution is supplied in the record.
• Literature checked at: 2026-08-21. This is supplied dated curation, not an independently verified assessment. It labels the problem open, reports proofs in small cases such as $p\leq6$, and describes the conjecture for all $p$ as unresolved. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, and sentence structure were repaired. These are supplied editorial claims; neither the external sources nor the mathematical formulation have been independently verified here.