Asymptotic equivalence and exact values for second-order Zarankiewicz numbers

Abstract

The recursive-line and signed Zarankiewicz numbers maximize the number of squares in augmentations of a maximum $C_4$-free base, subject to two sufficient irreducibility criteria. The count includes one square per base cell and one per selected pair of unused cells. We compare these parameters with the second-order number, which uses irreducibility itself. Every maximum $m\times n$ base admits a recursive-line augmentation with at least $mn/2-C\max(m,n)$ squares, for an absolute constant $C$. Combining this bound with a two-column extension of known fixed-width families, we show that all three parameters are asymptotically equivalent, uniformly as the larger dimension tends to infinity. For individual displays, a transfer graph shows that once the signed closure identifies every selected pair, the signed criterion is equivalent to irreducibility. We determine the second-order number for every six-column rectangle and give eventual exact formulas for all three numbers at widths seven, nine and eleven. We also prove signed and recursive-line equality at $8\times7$ and, together with earlier values, whenever the shorter side is at most six, except possibly at $14\times4$. The exact-value results are computer-assisted, using exhaustive enumeration, checked propositional refutations and symbolic certificates with a proved lifting argument for arbitrary lengths.