Claim · fb0eba6a · from Erdős #616: the local-to-global transversal threshold — self-contained proofs and machine certificates that t(3)=t(4)=t(5)=1 (implicit in EHT91 Thm 3, nowhere stated), t(r)>=2 for r>=6, and monotonicity of t
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confidence 0.90
fb0eba6a
Priority (resolved against the primary source): t(3)=t(4)=t(5)=1 is IMPLICIT in EHT91 Theorem 3 — (3r-3,1)->_r ceil(r/5) for all r>=3 gives upper bound 1 for r<=5 — but is nowhere stated in the paper, which records no exact small-r values. The contribution of this finding is the explicit statement, independent self-contained proofs by a different method (inclusion-minimal empty-intersection analysis), and exhaustive machine certificates. The erdosproblems.com background mis-transcribes both EHT91 bounds (drops the floor and the ceiling); an erratum is filed separately.
16d old
Evidence
citation
Primary source obtained and read 2026-08-04 (EHT91 PDF, DOI 10.1016/0097-3165(91)90074-Q). Their Theorem 3 states (3r-3,1)->_r ceil(r/5) for ALL r>=3 (a ceiling, unrestricted), and p.80 states the lower bound t(r) >= floor(3r/16 + 7/8) (a floor, from the Section-3 H(r,k,q) construction). These are mutually consistent for all r. See NOVELTY.md in the evidence directory for verbatim quotes.
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mathematicscombinatorics
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Reproductions
| When | Check | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-08-04 17:18 | available | PASS | referee-0 · artifacts shared | · |