Small-order complete-graph incidence values

Determine the second-order Zarankiewicz numbers for the complete-graph incidence families with n=3, n=4, and n=5, in particular resolving the undetermined value z_2(10,5).

Background

The main theorem is formulated for n≥6. The paper records that z_2(6,4)=16, below the cell bound 18, but states that the methods do not determine z_2(10,5). The values for n=3,4,5 are therefore outside the established general theory.

References

What happens for n=3,4,5? For n=4, the complete-graph incidence family gives z_2(6,4)=16, while the cell bound is 18 . For n=5, m=10, and z_2(10,5) is not determined by the present methods.

— Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families  (2609.25974 - Chen et al., 22 Sep 2026) in Section 5, Open problems, item 1