Triple

T29562853

Position	Surface form	Disambiguated ID	Type / Status
Subject	Turán's theorem	`E750082`	entity
Predicate	extremalGraph	`P148126`	FINISHED
Object	Turán graph T_{r-1}(n) The Turán graph \(T_{r-1}(n)\) is the complete \((r-1)\)-partite graph on \(n\) vertices with parts as equal in size as possible, which maximizes the number of edges among all \(K_r\)-free graphs on \(n\) vertices.	`E750082`	NE FINISHED

How this triple was built (3 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: Turán graph T_{r-1}(n) | Statement: [Turán's theorem, extremalGraph, Turán graph T_{r-1}(n)]

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: Turán graph T_{r-1}(n)
Triple: [Turán's theorem, extremalGraph, Turán graph T_{r-1}(n)]

Generated description

The Turán graph \(T_{r-1}(n)\) is the complete \((r-1)\)-partite graph on \(n\) vertices with parts as equal in size as possible, which maximizes the number of edges among all \(K_r\)-free graphs on \(n\) vertices.

PD Predicate disambiguation gpt-5-mini-2025-08-07

Target predicate: extremalGraph
Context triple: [Turán's theorem, extremalGraph, Turán graph T_{r-1}(n)]

A. isExtremal
Indicates that an entity represents an extreme (such as a maximum, minimum, or boundary) within a given set, structure, or ordering.
B. extremalFunction
Indicates that a function attains an extreme value (such as a minimum or maximum) under given conditions or over a specified domain.
C. extremalCase
Indicates a relationship where a situation, value, or configuration represents the most extreme or boundary case within a given set, range, or conditions.
D. isExtremalWhen
Indicates that a given entity or configuration attains an extreme value (such as a maximum or minimum) of some specified quantity or property under certain conditions.
E. typicalExtremalFamily chosen
Indicates that a given family of objects (such as sets or structures) exemplifies the standard or characteristic configuration that achieves an extremal bound in a particular combinatorial or optimization problem.
F. None of above.

Provenance (6 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_69f0bd4919e48190942b2a13d5b97d03`	completed	April 28, 2026, 1:59 p.m.
NER	Named-entity recognition	`batch_69f66d1e64e8819080579603e5bcdeb7`	completed	May 2, 2026, 9:31 p.m.
NED1	Entity disambiguation (via context triple)	`batch_6a262d67232c8190b8c5d890903aa257`	completed	June 8, 2026, 2:48 a.m.
NEDg	Description generation	`batch_6a2632e2296c8190996aae4a1c59a3af`	completed	June 8, 2026, 3:11 a.m.
NED2	Entity disambiguation (via description)	`batch_6a2636b244f481909a3625d8c3c87e01`	completed	June 8, 2026, 3:27 a.m.
PD	Predicate disambiguation	`batch_69f6633ac8a88190ab0cda62bbfcf9b0`	completed	May 2, 2026, 8:48 p.m.

Created at: April 28, 2026, 5:21 p.m.