A Computational Study of Limited Augmented Zarankiewicz Numbers in the Incidence-Graph Family of Complete Graphs

Let $G_1$ denote the incidence graph of the complete graph $K_{q+1}$. We study limited augmented Zarankiewicz numbers in this family by combining exact 0–1 ILP computations for the smallest cases with a constructive search procedure followed by exact admissibility verification in the larger cases considered here. We obtain $$z_L(6,4)=14,z_L(10,5)=26,z_L(15,6)\ge 43,z_L(21,7)\ge 64,z_L(28,8)\ge 88.$$ The first two values are exact. The three lower bounds arise from explicitly verified admissible families with $\mathrm{\mid }{E}_2\mathrm{\mid }=13$, $\mathrm{\mid }{E}_2\mathrm{\mid }=22$, and $\mathrm{\mid }{E}_2\mathrm{\mid }=32$, respectively; the families used to obtain these bounds are nondegenerate in the sense of [8]. In each case, the resulting value improves the corresponding classical Zarankiewicz number and hence strengthens the available lower bounds for BSR(m,n) within this family.