Maximum size of a three-sunflower-free family of four-element sets

The manuscript claims a computer-assisted exclusion of 69-member three-sunflower-free four-uniform families, giving 54 <= M(4,3) <= 68. It improves the upper bound 83 in Axante et al., arXiv:2609.06175v1, Theorem 6.9. The classical lower construction is reproduced, not claimed as new. The proof combines their intersecting-family bound 27, structural point-link reductions, Jin's graph-homomorphism theorem, exact certificates and finite exhaustive checks. A local cap of 61 requires a complete R16 point-link and is not a global bound. Cardinality 68 is not excluded, the exact value is not determined, and the general sunflower conjecture (Erdős #20) remains open.

The manuscript claims a computer-assisted exclusion of 69-member three-sunflower-free four-uniform families, giving 54 <= M(4,3) <= 68. It improves the upper bound 83 in Axante et al., arXiv:2609.06175v1, Theorem 6.9. The classical lower construction is reproduced, not claimed as new. The proof combines their intersecting-family bound 27, structural point-link reductions, Jin's graph-homomorphism theorem, exact certificates and finite exhaustive checks. A local cap of 61 requires a complete R16 point-link and is not a global bound. Cardinality 68 is not excluded, the exact value is not determined, and the general sunflower conjecture (Erdős #20) remains open.