Hole-count threshold for an overtake

Determine whether, for every pair of dimensions with min(m, n) ≥ 3, an overtake z2A(m, n) > z2(m, n) occurs if and only if the limited extremal configuration has at least five unoccupied cells, Hlimited(m, n) ≥ 5.

Background

An overtake is the strict inequality z2A(m, n) > z2(m, n), meaning that allowing a non-limited one-edge set improves the second-order Zarankiewicz bound. Theorem 9 shows that an overtake requires Hlimited(m, n) ≥ 3, while the known example (4, 4) has Hlimited = 5.

The authors conjecture that five holes are not merely necessary in the relevant range but also sufficient, apart from the degenerate width-two behavior explicitly excluded by the condition min(m, n) ≥ 3. The computed range supports the conjecture, with (4, 4) the only observed overtake.