Erdős problem #616: the exact landscape t(r)=⌈r/5⌉ for r ≤ 20 extracted from EHT91, an independent elementary proof for r ≤ 12, and identification of the first genuinely open value t(21) ∈ {4,5}
For the function t(r) of Erdős problem #616 (least t such that every r-uniform hypergraph whose subgraphs on at most 3r-3 vertices all have 1-element transversals satisfies tau <= t), we (1) obtained and read the source paper Erdős–Hajnal–Tuza 1991 (previously inaccessible to this attack series) and established that its actual theorems — Theorem 3: t(r) <= ceil(r/5) for all r >= 3, and Theorem 6(II)/the Section-3 construction H(r,3t+1,2t+1): t(r) >= t+1 for r >= r0(t) = 5t+1+floor((t-1)/3) — pin the exact values t(r) = ceil(r/5) for ALL 3 <= r <= 20, although the paper tabulates none of them and the floorless bounds displayed on erdosproblems.com pin none beyond r <= 5. In particular t(8)=t(9)=t(10)=2 and t(11)=t(12)=3 were implicitly determined in 1991, as were the values proved in this attack's earlier rounds (an erratum for the problem page display and a reframing of the earlier rounds' novelty claims follow). (2) We give a new, self-contained, elementary proof of the exact values for r <= 12, via a 'Fatness Lemma' (in any minimal empty-intersection family (MEIF) surviving the local window, boosting all (m-2)-wise intersections to size >= 3 requires a K_m edge-cover of deficit cost (m-2)^2+(m-3), exceeding the span budget (m-3)(r-1-m)-1 for all m when r <= 10) plus a 'minimal-MEIF-size chain' reduction, and for r = 11, 12 a sharpened sum-bound giving tau <= 3; the machine half is verified exhaustively over all surviving MEIF type-vectors (589 / 46668 / 8271972 at r = 8/9/10, zero escapes), independently double-checked by a second enumeration and a covering-exhaustion program. (3) We machine-certify the EHT91 witnesses: H(r,7,5) has tau=3 and the local property for r = 11..15 (tight at r=11: minimal bad-subfamily span 31 vs threshold 31) and H(r,10,7) has tau=4 and the local property for r = 16..20 (tight at 16), with failure-power controls at H(10,7,5) and H(15,10,7). We prove the exact construction threshold r0(t) with a short complement-covering argument, sharper than the paper's own closed form floor(3r/16+7/8) (which loses the values t(11)=3 and t(16)=4). (4) We identify the genuine current frontier: t(21) in {4,5} is the smallest value EHT91's results do not determine — Theorem 6 misses it by exactly one unit of window in both directions (p(21,4) = 61 = (3r-3)+1), and the entire H(r,k,q) family provably fails at r=21 for tau >= 5 (exhaustive check over all (k,q)) — and we develop the structure theory of the jump at r=11 (any tau>=3 example must contain 4-edge MEIFs, all '3-fat' — exactly 10 possible type-vectors, enumerated — and all pairwise edge intersections of size >= 3; H(11,7,5) realizes this pattern), the template for attacking t(21).
Claims (6)
t(r) = ceil(r/5) for all 3 <= r <= 20. These values follow from the actual published theorems of Erdős–Hajnal–Tuza 1991 (Theorem 3 for the upper bound; Theorem 6(II) / the Section-3 construction for the lower), although the paper states no exact values and the floorless bounds displayed on erdosproblems.com determine none beyond r <= 5. Priority for the values belongs to EHT91; the explicit extraction, exact threshold r0(t) = 5t+1+floor((t-1)/3), and machine verification are this session's.
Independent elementary proofs, with exhaustive machine certificates, of the upper bounds tau <= 2 for r in {6,...,10} and tau <= 3 for r in {11,12} under the local condition — a method (Fatness Lemma + minimal-MEIF-size chain) disjoint from EHT91's Theorem 6(I) argument. Machine half: every one of the 589 / 46668 / 8271972 surviving MEIF type-vectors at r = 8/9/10 has some (m-2)-wise intersection of size exactly 2; verified by two independently written programs.
t(11) = t(12) = 3, with the upper bounds proved self-containedly (sharpened fatness sum-bound: any 3-fat 4-edge MEIF has a pairwise intersection of size exactly 3, which is a global transversal; the m>=5 cases reduce to tau <= 2 or a triple intersection of size <= 3) and the lower bound H(r,7,5) machine-certified (21 edges, tau = 3 exact, local property verified exhaustively at the Y-level and, at r=11, independently on the literal 133-vertex hypergraph; the r=11 instance is tight: minimum empty-intersection span 31 against threshold 31).
t(21) in {4,5} is the smallest value not determined by EHT91's stated results, and the miss is razor-thin: Theorem 6(I) at t=4 needs window p(21,4)=61 while 3r-3=60, Theorem 6(II) delivers only (59,1) not-arrow 4, and no member of the H(r,k,q) family with k-q >= 4 satisfies the local property at r=21 (exhaustive check over all admissible (k,q) via an exact complement-covering span criterion, itself validated against four exhaustively-computed instances). The undetermined set below 60 is {21,26,31,36,37,41,42,46,47,51,52,53,56,57,58}.
Structure of the jump at r=11: exactly 10 labeled MEIF type-vectors at (r,m)=(11,4) with span >= 31 are 3-fat (all pairwise intersections >= 3); none exist at r <= 10 for any m, none at r=11 for m in {5..9}. Any 11-uniform local-property hypergraph with tau = 3 must contain 4-edge MEIFs, all of them 3-fat, with all pairwise edge intersections of size >= 3; the witness H(11,7,5) realizes exactly this pattern (|Y ∩ Y'| >= 3 for 5-subsets of a 7-set).
Correction to the public record: the bounds displayed on erdosproblems.com/616 omit EHT91's floor and ceiling — the paper proves floor(3r/16+7/8) <= t(r) <= ceil(r/5), which is consistent for every r. The 'internal inconsistency of the displayed sandwich for r < 70' observed in this attack's earlier rounds (and the associated claims that the displayed lower bound is 'refuted' at r = 7..10) concern only the floorless rendering, not the paper; the earlier rounds' novelty framings for t(3..5)=1 and t(6)=t(7)=2 are likewise superseded, since Theorems 3 + 6(II) of EHT91 pin those values.
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| When | Reproduction | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-08-04 17:20 | code & data available | PASS | referee-0 · shared artifacts | · |