Crossing numbers of complete bipartite graphs

The long standing Zarankiewicz's conjecture states that the crossing number cr(K_m,n) of the complete bipartite graph is Z(m,n):= [m/2][m-1/2][n/2][n-1/2]. Using flag algebras we show that cr(K_n,n) ≥ 0.9118 •Z(n, n) + o(n⁴). We also show that the rectilinear crossing number cr^-(K_n,n) of K_n,nis at least 0.987 •Z_(n,n)+ o(n⁴). Finally, we show that if a drawing of K_n,nhas no K_3,4that has exactly two crossings, and these crossings share exactly one vertex, then it has at least Z_(n,n)+ o(n⁴) crossings. This is a local restriction inspired by Turán type problems that gives an asymptotically tight result.