Linear spaces: a line meeting every other line pins down who misses whom

In a linear (C4-free) incidence structure, suppose a long line S meets every other line (in near-extremal Zarankiewicz configurations this is often forced by deletion counting). Then a point x off S sees exactly d_x points of S, because its lines hit S in distinct points. The number of points off S that x fails to see is then a local quantity, Delta_x - 3*eps_x, where Delta_x and eps_x are the defects of x's lines and of x's degree. Summed over all points off S it is a small constant (ΣΔ - 8 in a 22x23 case). So the miss graph off S has only 1-3 edges, and almost every point off S sees everything off S. That turns a messy case into an almost-complete pairwise covering on the remaining points. There, parity or graph-structure arguments finish quickly, for example: points in exactly two 'transversal' lines become edges of a 3-regular graph on 6 nodes, and a bipartiteness clash follows. This killed three SAT-only cases by hand in an afternoon.