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

For Erdős problem #616 (r-uniform hypergraphs in which every induced subgraph on at most 3r-3 vertices has transversal number <= 1; determine the best-possible global bound t(r) on tau), we prove with complete elementary proofs and machine-checked certificates: (1) t(3)=t(4)=t(5)=1 — for r <= 5 the local condition forces a global common vertex, since every inclusion-minimal empty-intersection family of m r-sets spans at most m(r-m+2) <= 3r-3 vertices and therefore sits inside a forbidden window; (2) t(r) >= 2 for every r >= 6, via an explicit 4-edge gadget on 4r-8 vertices (three-wise intersecting, empty total intersection, span 4r-8 > 3r-3) whose local property is verified both by a proved subfamily criterion and by exhaustive enumeration of all vertex subsets at r=6,7; (3) t(r+1) >= t(r) for all r >= 3, via a pendant-vertex extension that preserves both the local property and tau exactly. Together these locate the exact threshold r=6 where the local condition stops forcing tau=1. The results are consistent with the Erdős–Hajnal–Tuza (1991) bounds displayed at erdosproblems.com (their lower bound equals 2 exactly at r=6); the displayed EHT bounds cannot be literally valid at small r (at r=5 they read 1.8125 <= t <= 1), and no record of the exact small-r values was found on the problem page, its forum, or in searches — though EHT91 itself is paywalled and could not be read, so priority over EHT91 is explicitly unconfirmed. The correct fragments of the retracted Jan 2026 ChatGPT 5.2 Pro attempt (posted by forum user jkabrg; refuted in Step 4 by comments of TerenceTao and Nat Sothanaphan) are re-derived independently and credited; nothing from its erroneous upper-bound claim t(r)=2 for all r>=6 is used. [Priority update 2026-08-04: EHT91 was subsequently obtained and read; the small-r values are implicit in (but nowhere stated by) its Theorem 3, whose true form has a ceiling: t(r) <= ceil(r/5) for all r>=3. This finding supplies the first explicit statement, independent proofs by different methods, and machine certificates.] Landscape correction (t8/t11 round, 2026-08-04): reading EHT91's Theorem 6(II) directly — not the lossy closed form floor(3r/16+7/8), which hides it — the paper's own results determine t(r)=ceil(r/5) for ALL 3<=r<=20 (in particular t(11)=t(12)=3); the first value EHT91 leaves open is t(21) in {4,5}. No exact value below r=21 is new; this finding's proofs stand as independent verification with new machinery and certificates.