Similarly, place vertices on the -axis of the plane, avoiding the origin, with equal or nearly-equal numbers of points above and below the -axis. Then, connect every point on the -axis by a straight line segment to every point on the -axis. However, their proofs that this formula is optimal, that is, that there can be no drawings with fewer crossings, were erroneous. The gap was not discovered until eleven years after publication, nearly simultaneously by Gerhard Ringel and Paul Kainen. Nevertheless, it is conjectured that Zarankiewicz's and Urbanik's formula is optimal. This has come to be known as the Zarankiewicz crossing number conjecture. Although some special cases of it are known to be true, the general case remains open. Since ' and ' are isomorphic, it is enough to consider the case where "". In addition, for Zarankiewicz's construction gives no crossings, which of course cannot be bested. So the only nontrivial cases are those for which and are both . Zarankiewicz's attempted proof of the conjecture, although invalid for the general case of , works for the case . It has since been extended to other small values of , and the Zarankiewicz conjecture is known to be true for the complete bipartite graphs with . The conjecture is also known to be true for , , and . If a counterexample exists, that is, a graph requiring fewer crossings than the Zarankiewicz bound, then in the smallest counterexample both and must be odd. For each fixed choice of , the truth of the conjecture for all can be verified by testing only a finite number of choices of . More generally, it has been proven that every complete bipartite graph requires a number of crossings that is (for sufficiently large graphs) at least 83% of the number given by the Zarankiewicz bound. Closing the gap between this lower bound and the upper bound remains an open problem. If edges are required to be drawn as straight line segments, rather than arbitrary curves, then some graphs need more crossings than they would when drawn with curved edges. However, the upper bound established by Zarankiewicz for the crossing numbers of complete bipartite graphs can be achieved using only straight edges. Therefore, if the Zarankiewicz conjecture is correct, then the complete bipartite graphs have rectilinear crossing numbers equal to their crossing numbers.