Crossing numbers of complete bipartite graphs

The long standing Zarankiewicz's conjecture states that the crossing number cr(K-m,K-n) of the complete bipartite graph is Z(m,n) := left perpendicular m/2right perpendicular left perpendicular m-1/2 right perpendicular left perpendicular n/2 right perpendicular left perpendicular n-1/2 right perpendicular. Using flag algebras we show that cr(K-n,K-n) >= 0.9118 Z(n, n) + o(n(4)). We also show that the rectilinear crossing number (cr) over bar (K-n,K-n) of K-n,K-n is at least 0.987 . Z(n,n) + o(n(4)). Finally, we show that if a drawing of K-n,K-n has no K-3,K-4 that has exactly two crossings, and these crossings share exactly one vertex, then it has at least Z(n,n) + o(n(4)) crossings. This is a local restriction inspired by Turan type problems that gives an asymptotically tight result. (C) 2023 The Authors. Published by Elsevier B.V.