Zarankiewicz's crossing-number conjecture (Turan's brick factory problem)

Claims $\mathrm{cr}(K_{m,n})=\lfloor\frac m2\rfloor\lfloor\frac{m-1}2\rfloor\lfloor\frac n2\rfloor\lfloor\frac{n-1}2\rfloor$ for all positive $m,n$. The lower bound comes from a linear-algebra inequality $\sum_{i,j}\dim(U_i\cap V_j)\le\lfloor m^2/4\rfloor N$ for subspaces with $U_i\cap V_i=0$, proved by interpolation, applied through signed intersections of cycles; Zarankiewicz's drawing gives the upper bound. Crossings are counted as points. It does NOT address rectilinear, pair or odd crossing numbers of $K_{m,n}$.

Claims $\mathrm{cr}(K_{m,n})=\lfloor\frac m2\rfloor\lfloor\frac{m-1}2\rfloor\lfloor\frac n2\rfloor\lfloor\frac{n-1}2\rfloor$ for all positive $m,n$. The lower bound comes from a linear-algebra inequality $\sum_{i,j}\dim(U_i\cap V_j)\le\lfloor m^2/4\rfloor N$ for subspaces with $U_i\cap V_i=0$, proved by interpolation, applied through signed intersections of cycles; Zarankiewicz's drawing gives the upper bound. Crossings are counted as points. It does NOT address rectilinear, pair or odd crossing numbers of $K_{m,n}$.