Sharp hole budget for the second-order augmented Zarankiewicz number

Prove that the second-order augmented Zarankiewicz number satisfies z2A(m, n) ≤ z2(m, n) + ⌊Hlimited(m, n)/3⌋ for all m, n ≥ 2, equivalently establishing the corresponding lower bound on the combined number of unoccupied cells and one-edge deficit in every configuration counted by z2A(m, n).

Background

The quantity Hlimited(m, n) denotes the number of unoccupied cells in a limited extremal configuration. Theorem 9 proves the weaker upper bound z2A(m, n) ≤ z2(m, n) + ⌊(Hlimited(m, n) − 1)/2⌋ and shows that any overtake requires at least three limited holes.

The conjectured bound reflects the authors’ observation that increasing the displayed length beyond z2 should require a cost of three holes per unit of excess. The bound is known to be sharp at (m, n) = (4, 4), where Hlimited = 5 and z2A − z2 = 1, but it is not proved in general.