Even-p row pairing and certification

Construct, for every even p, a fixed-point-free pairing of the p rows in the near-perfect one-factorization construction for n=2p+1 that replaces the involution t↦2a−t, and determine whether the resulting configuration satisfies the recursive-line criterion (RW3+).

Background

For odd p, the intra-factor pairing uses the involution t↦2a−t and the resulting construction is certified. For even p, that involution has a fixed point, so the construction has the correct rank but lacks the transfer certificate. The cases p=4 and p=6 are settled by configurations of different shapes, while the general even-p branch remains unresolved.

References

Which fixed-point-free pairing of the p rows replaces the involution t\mapsto2a-t, and does the resulting configuration satisfy (\mathrm{RW}3+)? More generally, is there a systematic even-p family attaining Z(2p+1), or is the star-deletion-and-repair operation the only route, and does it then succeed at every even p starting from the certified construction of order 2p+3?

— Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families  (2609.25974 - Chen et al., 22 Sep 2026) in Section 5, Open problems, item 2 (Open Problem~\ref{op:evenp})

What is missing is a theory. Is the re-pairing of the orphaned cells always possible, and if so, is there a canonical choice of it in place of the randomized local search used here? Which of the remaining orders of Conjecture~\ref{conj:all-n} can be reached this way, for instance n=18 and n=17, starting from the certified odd order n=19 (p=9), and n=16, starting from n=17?

— Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families  (2609.25974 - Chen et al., 22 Sep 2026) in Section 5, Open problems, item 4