Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: König's lemma in infinite graph theory | Statement: [Julius König, knownFor, König's lemma in infinite graph theory]

Target entity: König's lemma in infinite graph theory
Context triple: [Julius König, knownFor, König's lemma in infinite graph theory]

• A. Kruskal's tree theorem
  Kruskal's tree theorem is a fundamental result in combinatorics and mathematical logic stating that finite trees are well-quasi-ordered under homeomorphic embedding, with deep implications in proof theory and computer science.
• B. Ky Fan’s lemma
  Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
• C. Menger theorem in graph theory
  Menger's theorem in graph theory is a fundamental result that characterizes the connectivity between two vertices in a graph by equating the maximum number of pairwise internally disjoint paths between them with the minimum size of a vertex cut separating them.
• D. Tarski’s fixed point theorem
  Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.
• E. Kuratowski’s theorem on planar graphs
  Kuratowski’s theorem on planar graphs is a fundamental result in graph theory that characterizes planar graphs by stating that a finite graph is planar if and only if it contains no subgraph that is a subdivision of the complete graph K₅ or the complete bipartite graph K₃,₃.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: König's lemma in infinite graph theory
Target entity description: König's lemma in infinite graph theory is a fundamental result stating that every infinite, finitely branching tree contains an infinite path.

• A. Kruskal's tree theorem
  Kruskal's tree theorem is a fundamental result in combinatorics and mathematical logic stating that finite trees are well-quasi-ordered under homeomorphic embedding, with deep implications in proof theory and computer science.
• B. Ky Fan’s lemma
  Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
• C. Menger theorem in graph theory
  Menger's theorem in graph theory is a fundamental result that characterizes the connectivity between two vertices in a graph by equating the maximum number of pairwise internally disjoint paths between them with the minimum size of a vertex cut separating them.
• D. Tarski’s fixed point theorem
  Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.
• E. Kuratowski’s theorem on planar graphs
  Kuratowski’s theorem on planar graphs is a fundamental result in graph theory that characterizes planar graphs by stating that a finite graph is planar if and only if it contains no subgraph that is a subdivision of the complete graph K₅ or the complete bipartite graph K₃,₃.
• F. None of above. chosen