Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-11T22:09:04.139151Z
Paper Citation Record · LEDGER
Frequent elements in union-closed set families
As of 16 August 2026, this Paper Citation Record lists 22 of 22 outbound references and 2 inbound Pith citation observations for arXiv:2412.03862.
A citation records a reference. It does not transfer a finding from one paper to another.
Coverage vector
One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-16T06:30:59.297886+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links, observed 2026-08-11T22:11:03.023621Z
A source-named dated measurement, never combined with another source.
Source: pith, observed 2026-08-06T17:34:16.351122Z
Reference resolution
22 of 22 outbound references displayed
- verified exact1
- verified fuzzy4
- unresolved17
- parse uncertain0
- malformed identifier0
- metadata mismatch0
External citation measurements
No source-named external measurement is stored.
Outbound references
Observation e257eff2-dbe5-443f-a002-a3d28956280b · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 1
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.734862Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation b21d5b58-8e22-4eb2-ae15-e31e503fecb4 · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 2
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.709135Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation b548d57e-983f-4849-a235-eb15a5c0912f · outbound
This paper cites Minimum density of union-closed families.
Frequent elements in union-closed set families Minimum density of union-closed families
Reference 3
- Resolution
- verified exact
- local_arxiv, observed 2026-08-11T22:09:04.368959Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation f1541ce8-ff72-4f04-9b19-5b2c7fef47c7 · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 4
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.686024Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation d03a0898-649b-4d70-b19a-fc028489c06f · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 5
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.659486Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 7f8a75ed-1823-405c-b224-37d73685f6c3 · outbound
This paper cites 31 (2015), no.
Frequent elements in union-closed set families 31 (2015), no
Reference 6
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T22:09:04.635016Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation eba2fac8-fc24-423e-9eb9-91e0acbf46ab · outbound
This paper cites Better bounds for the union-closed sets conjecture using the entropy approach.
Frequent elements in union-closed set families Better bounds for the union-closed sets conjecture using the entropy approach
Reference 7
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T22:09:04.016595Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 9eaa1e48-4c1a-49e3-9171-96a926defcc9 · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 8
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.612160Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation b6a324c9-4f08-485a-b825-0adef542d99e · outbound
This paper cites A constant lower bound for the union-closed sets conjecture.
Frequent elements in union-closed set families A constant lower bound for the union-closed sets conjecture
Reference 9
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T22:09:04.032512Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation cda59be6-2024-4b1e-b03a-207e0e09fa7e · outbound
This paper cites Graph generated union-closed families of sets.
Frequent elements in union-closed set families Graph generated union-closed families of sets
Reference 10
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T22:09:04.039940Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 84482078-5ee1-4ec7-96bd-6c53e2af9c51 · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 11
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.589777Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 4ba36ff3-b2d9-4677-b664-db62c377ebc1 · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 12
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.564965Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation ab290f20-d72a-42fa-a501-7c907739b9a9 · outbound
This paper cites Notes on the Union Closed Sets Conjecture.
Frequent elements in union-closed set families Notes on the Union Closed Sets Conjecture
Reference 13
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T22:09:04.063445Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 6f846439-0bf6-46bd-abe8-3202f299f23d · outbound
This paper cites Extension of a Method of Gilmer.
Frequent elements in union-closed set families Extension of a Method of Gilmer
Reference 14
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T22:09:04.072624Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation fca4b1e8-26f3-452e-b492-c6c2cd4f3656 · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 15
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.541456Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation b7830b16-63bc-4857-b17c-fdec61dc618e · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 16
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.520995Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 7f4689ca-db12-4b27-b1df-813719b0a180 · outbound
This paper cites 16 (2000), no.
Frequent elements in union-closed set families 16 (2000), no
Reference 17
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T22:09:04.495095Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 55d81f2f-1be4-4025-a26f-d6a26264b6dd · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 18
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.466751Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 92f1ea0b-0472-45e4-8383-e9780e3fa5a8 · outbound
This paper cites An improved lower bound for the union-closed set conjecture.
Frequent elements in union-closed set families An improved lower bound for the union-closed set conjecture
Reference 19
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T22:09:04.118904Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation db33b8f6-2f90-4d5c-9ed9-9047852a92e1 · outbound
This paper cites Vaughan, Families implying the F rankl conjecture , European J.
Frequent elements in union-closed set families Vaughan, Families implying the F rankl conjecture , European J
Reference 20
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T22:09:04.444461Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation e4b07fba-fe52-41ef-a3b7-9037b6da67c3 · outbound
This paper cites an unresolved cited work.
Frequent elements in union-closed set families Unresolved cited work
Reference 21
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T22:09:04.417947Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation a131c50a-07ab-4961-a10a-74b9a46de6df · outbound
This paper cites 199 (1999), no.
Frequent elements in union-closed set families 199 (1999), no
Reference 22
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T22:09:04.391746Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Pith citing papers
Observation 04cc6297-2a89-4ad3-9543-70545d925e95 · inbound
Further analysis on the second frequency of union-closed set families cites this paper.
Further analysis on the second frequency of union-closed set families Frequent elements in union-closed set families
Reference 1
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T22:11:03.023621Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 20103597-3ef3-4881-b74d-be5a415196fd · inbound
A lemma on a finite union-closed family of finite sets and its applications cites this paper.
A lemma on a finite union-closed family of finite sets and its applications Frequent elements in union-closed set families
Reference 4
- Resolution
- verified exact
- local_arxiv, observed 2026-08-06T17:34:16.356354Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.