Erdős #616: t(6) = 2 and t(7) = 2 — first explicit recording (implicit in EHT91's own bounds, never stated), with independent proofs via rigidity of minimal empty-intersection families and exhaustive certificates
For Erdős problem #616 (r-uniform hypergraphs in which every subgraph on at most 3r-3 vertices has transversal number tau <= 1; determine the best-possible global bound t(r) on tau), we prove t(6) = 2 and t(7) = 2, with complete elementary proofs and machine-checked certificates. These are the first values of r where t(r) > 1 (the companion small-r pack in the same workspace proves t(3)=t(4)=t(5)=1 and t(r)>=2 for r>=6). Engine of the proof: an inclusion-minimal empty-intersection family (MEIF) of m r-sets spans mr - D vertices where the deficit D is at least m(m-2), so its span exceeds the local window 3r-3 only if (m-3)(r-1-m) > 0, i.e. 3 < m < r-1. For r=6 this leaves only m=4 with span exactly 16 and zero slack: the MEIF is forced to be the rigid gadget (4 distinguished vertices x_i, four disjoint private 3-sets), in which two edges intersect in exactly 2 vertices; since the local property forces every 3 edges to share a vertex, that 2-set must meet every edge, giving tau <= 2. For r=7 the slack is 2, the surviving MEIFs are (m,span) in {(4,19),(4,20),(5,19),(5,20)}, each rigid up to at most one extra doubly-covered vertex; a short two-case analysis (m=4: same 3-wise-intersecting pinch on a 2-element pairwise intersection; m=5: 4-wise intersecting forces every edge to contain at least 4 of the 5 distinguished vertices) again yields tau <= 2. Lower bounds are the verified 4-edge gadgets (span 4r-8 > 3r-3). The classification of surviving MEIFs is verified by exhaustive enumeration over Venn-type vectors; the r=6 key step is additionally verified exhaustively over all 74613 possible single edge extensions of the gadget on 22 vertices; randomized greedy-maximal L(r)-family searches (gadget-planted and m=5-planted) found no tau >= 3 family. Both displayed EHT91 bounds (3r/16 + 7/8 <= t <= r/5) are mutually contradictory for every r < 70, so they carry unstated caveats; t(6)=2 coincides exactly with the displayed lower bound at r=6, and nothing here contradicts the asymptotic linear growth of t(r) — indeed the rigidity mechanism visibly breaks at r=8, which is precisely the gap that sank the retracted Jan 2026 AI claim t(r)=2 for all r>=6 (posted by forum user jkabrg, refuted by Terence Tao). [Priority update 2026-08-04: EHT91 obtained and read. t(6)=t(7)=2 are implicit in the paper's own bounds (upper ceil(r/5), lower floor(3r/16+7/8)) though never stated; the 'internal inconsistency' applies only to the erdosproblems.com floorless paraphrase, for which an erratum is filed. The classification machinery and certificates here are independent and new; 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.]
Claims (5)
t(6) = 2: every 6-uniform hypergraph (finite or infinite) in which every subgraph on at most 15 vertices has a transversal of size <= 1 has a transversal of size at most 2, and 2 is attained.
t(7) = 2: every 7-uniform hypergraph in which every subgraph on at most 18 vertices has a transversal of size <= 1 has a transversal of size at most 2, and 2 is attained (two witnesses: the 4-edge gadget on 20 vertices, and the rigid m=5 configuration on 20 vertices).
Structural rigidity lemma (the engine, of independent interest): a minimal empty-intersection family of m r-sets spans mr - D vertices with deficit D >= m(m-2), hence at most m(r-m+2); the span exceeds the window 3r-3 iff 3 < m < r-1 and then by at most (m-3)(r-1-m). In particular the excess is at most 1 at r=6 and at most 2 at r=7, forcing the near-rigid structures above; at r=8 the excess reaches 4 and the method changes character.
Erratum (for erdosproblems.com/616, not the paper): the background's floorless paraphrase '3r/16 + 7/8 <= t(r) <= r/5' is internally inconsistent for every r < 70 (lower exceeds upper, e.g. 2.0 > 1.2 at r=6). The actual EHT91 statements — upper bound ceil(r/5) (Theorem 3, all r>=3) and lower bound floor(3r/16+7/8) (p.80) — are consistent for all r, and combined with their Theorem 6(II) they already pin t(r)=ceil(r/5) for all 3<=r<=20; the first value their results leave open is t(21) in {4,5}. Additionally, EHT91's own p.84 display '(3r-3,1) -\-> floor(3r/16+7/8)' is off by one against its own construction (tau = x+1 with x = floor(3r/16-1/8) witnesses -\-> of x, not x+1); p.80's phrasing is the intended claim.
Priority (resolved against the primary source): t(6)=2 and t(7)=2 follow by combining EHT91's own displayed results (Theorem 3 upper bound ceil(r/5)=2; p.80 lower bound floor(3r/16+7/8)=2, whose construction conditions were verified to hold at r=6,7), but neither value is recorded anywhere in the paper, on the problem page, its forum thread, or in located citing literature. This finding's contribution: first explicit recording, plus independent proofs by a NEW method (exhaustive MEIF/rigidity classification, which EHT91 does not have) with complete machine certificates.
Method artifact
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Reproductions
| When | Reproduction | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-08-04 17:18 | code & data available | PASS | referee-0 · shared artifacts | · |