Extremal Bipartite Containment of B-t

Definitions and mathematical statement

For a positive integer parameter $t$, define the bipartite graph $B_t$ with disjoint vertex classes \[ L=\{x_i:1\le i\le t\}\sqcup\{y_j:1\le j\le t\}\sqcup\{z_k:1\le k\le t\},\qquad R=\{w_{ijk}:1\le i,j,k\le t\}, \] where all listed vertices are distinct, and with edge set \[ E(B_t)=\left\{\{x_i,w_{ijk}\},\{y_j,w_{ijk}\},\{z_k,w_{ijk}\}:1\le i,j,k\le t\right\}. \] Under the ordinary subgraph interpretation, a graph $G$ contains a copy of $B_t$ if there is an injective map $\phi:V(B_t)\to V(G)$ such that $\{\phi(u),\phi(v)\}\in E(G)$ whenever $\{u,v\}\in E(B_t)$; additional edges between image vertices are permitted.

The supplied question asks whether there exist real constants $\sigma<3$ and $C>0$ such that every sufficiently large bipartite graph $G=(A\sqcup B,E)$ satisfies \[ \left(|A|^\sigma<|B|\ \text{ and }\ |E|>C|B|\right) \quad\Longrightarrow\quad G\text{ contains a copy of }B_t. \] This is an incompletely qualified question: the record does not specify the quantifier over $t$, the permitted dependence of the constants and size threshold on $t$, or which size parameter must be sufficiently large. It also does not explicitly specify finite simple host graphs, induced versus ordinary containment, or whether a copy must send $L$ into $A$ and $R$ into $B$. The definition of a copy above explains the ordinary subgraph reading, without resolving these source qualifications. The intended domain must also make $|A|^\sigma$ defined; the record does not explicitly exclude $A=\varnothing$.

The supplied record labels the statement corrected_verified and says it was reconstructed and checked against the official report. Its supplied dated literature assessment nevertheless calls the extracted question malformed. These conflicting qualifications remain unresolved here.

Short English statement

The question asks whether a sufficiently unbalanced bipartite graph, with more than a constant times $|B|$ edges, must contain the graph encoding all triples from three disjoint sets of $t$ vertices. The quantification of $t$, the size threshold, and the containment conventions need confirmation. The supplied open status and verification claims are curation metadata requiring review.

Sources

• Local UnsolvedMath registry, permanent ID $30005816$, title: Extremal Bipartite Containment of B-t; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and supplies proposed year $2024$; no individual proposer is named.
• Supplied source citation: Discrete Geometry (2024), Oberwolfach Reports, pp. 137--202, DOI 10.4171/owr/2024/3. Supplied link: https://doi.org/10.4171/owr/2024/3.
• Supplied literature source: Oberwolfach Report 3/2024, https://ems.press/content/serial-article-files/48651.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It labels the problem open, describes the extracted question as malformed, says the report presents it as an open extremal problem, and reports that no authoritative resolution of the stated claim was found. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The record separately labels the statement corrected_verified and states that it was reconstructed and checked against the official Oberwolfach report. This supplied verification claim has not been independently checked here.
• The original extracted passage additionally mentions $f(n)$, described as the minimum of an undefined $f(X)$ over all $n$-element planar sets $X$ in general position, and contains the incomplete reference fragment: Erdős, On some unsolved problems in elementary geometry (in Hungarian), Mat. Its connection to the bipartite question is unspecified; it does not establish attribution of that question to Erdős.