Paper Citation Record · LEDGER

Frequent elements in union-closed set families

As of 14 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.

pith.paper-citation-record.v1
2412.03862 v3

Coverage vector

measured 22 of 22 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-11T22:09:04.139151Z

measured 24 of 24 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-14T06:32:32.682623+00:00

measured 2 of 2 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-11T22:11:03.023621Z

measured 0 of 1 external citation measurements

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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+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 2026-08-14T06:32:32.682623+00:00.