Triple

T8657127
Position Surface form Disambiguated ID Type / Status
Subject Ronald L. Graham E205446 entity
Predicate notableIdea P4 FINISHED
Object Graham–Pollak theorem
The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
E748752 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
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).
Input
Phrase: Graham–Pollak theorem | Statement: [Ronald L. Graham, notableIdea, Graham–Pollak theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Graham–Pollak theorem
Context triple: [Ronald L. Graham, notableIdea, Graham–Pollak theorem]
  • A. Erdős–Ko–Rado theorem
    The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
  • B. Erdős–Gallai theorem
    The Erdős–Gallai theorem is a fundamental result in graph theory that characterizes which sequences of nonnegative integers can occur as the degree sequences of simple graphs.
  • C. Birkhoff–von Neumann theorem
    The Birkhoff–von Neumann theorem is a fundamental result in matrix theory and combinatorics stating that every doubly stochastic matrix can be expressed as a convex combination of permutation matrices.
  • D. de Bruijn–Erdős theorem
    The de Bruijn–Erdős theorem is a fundamental result in combinatorics and graph theory that relates finite and infinite structures, notably asserting that certain properties of infinite graphs or set systems are determined by their finite substructures.
  • E. Erdős–Stone theorem
    The Erdős–Stone theorem is a fundamental result in extremal graph theory that asymptotically determines the maximum number of edges in an n-vertex graph that avoids containing a given subgraph.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Graham–Pollak theorem
Triple: [Ronald L. Graham, notableIdea, Graham–Pollak theorem]
Generated description
The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Graham–Pollak theorem
Target entity description: The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
  • A. Erdős–Ko–Rado theorem
    The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
  • B. Erdős–Gallai theorem
    The Erdős–Gallai theorem is a fundamental result in graph theory that characterizes which sequences of nonnegative integers can occur as the degree sequences of simple graphs.
  • C. Birkhoff–von Neumann theorem
    The Birkhoff–von Neumann theorem is a fundamental result in matrix theory and combinatorics stating that every doubly stochastic matrix can be expressed as a convex combination of permutation matrices.
  • D. de Bruijn–Erdős theorem
    The de Bruijn–Erdős theorem is a fundamental result in combinatorics and graph theory that relates finite and infinite structures, notably asserting that certain properties of infinite graphs or set systems are determined by their finite substructures.
  • E. Erdős–Stone theorem
    The Erdős–Stone theorem is a fundamental result in extremal graph theory that asymptotically determines the maximum number of edges in an n-vertex graph that avoids containing a given subgraph.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69ca8350897c819086cde7596fbe5fe7 completed March 30, 2026, 2:06 p.m.
NER Named-entity recognition batch_69cc484569788190aa41395854684e6f completed March 31, 2026, 10:18 p.m.
NED1 Entity disambiguation (via context triple) batch_69ceccec941881908263cd3205f10ccd completed April 2, 2026, 8:09 p.m.
NEDg Description generation batch_69cece8c4bdc8190988990c675f50f86 completed April 2, 2026, 8:16 p.m.
NED2 Entity disambiguation (via description) batch_69cecf3a0e78819082cc7c43eceae309 completed April 2, 2026, 8:19 p.m.
Created at: March 30, 2026, 6:30 p.m.