Density and separation for augmented Zarankiewicz numbers

Abstract

We study the augmented Zarankiewicz problem, in which disjoint pairs of cells are added to a binary matrix with no all-one $2\times2$ submatrix. The pairs must satisfy compatibility conditions, and the objective counts each original occupied cell and each added pair once. We show that starting with a maximum $C_4$-free matrix can lower the final optimum, answering a question of Qi, Cui, and Xu. Let ${z_A}(m,n)$ be the optimum over all $C_4$-free initial matrices, and ${z_L}(m,n)$ the optimum when the initial matrix must have the maximum number of occupied cells. As $n\to\infty$ with $n\le m=o(n^2)$, we prove \[ {z_A}(m,n)-{z_L}(m,n)\ge\left(\frac1{30}-o(1)\right)mn \] and determine the sharp second-order term: \[ {z_A}(m,n)=\frac{mn}{3}+\left(\frac1{\sqrt6}+o(1)\right)n\sqrt m. \] An explicit construction gives a separation at $m=n=1893$. We also find a sharp density threshold: when $n\to\infty$ and $m/n^2\to c>0$, the limited density ${z_L}(m,n)/(mn)$ tends to $1/3$ if and only if $c\ge1/12$. The proofs combine density and stability estimates, combinatorial constructions, and an exact polynomial certificate.