the following question: is it the case that for any fixed and , we have ? In the special case , up to constant factors, has the same order as , the maximum number of edges in an -vertex (not necessarily bipartite) graph that has no as a subgraph. In one direction, a bipartite graph with vertices on each side and edges must have a subgraph with vertices and at least edges; this can be seen from choosing vertices uniformly at random from each side, and taking the expectation. In the other direction, we can transform a graph with vertices and no copy of into a bipartite graph with vertices on each side of its bipartition, twice as many edges and still no copy of , by taking its bipartite double cover. Same as above, with the convention that , it has been conjectured that for all constant values of . For some specific values of (e.g., for sufficiently larger than , or for ), the above statements have been proved using various algebraic and random algebraic constructions. At the same time, the answer to the general question is still unknown to us. Incidence graphs in finite geometry For , a bipartite graph with vertices on each side, edges, and no may be obtained as the Levi graph, or point-line incidence graph, of a projective plane of order , a system of points and lines in which each two points determine a unique line, and each two lines intersect at a unique point. We construct a bipartite graph associated to this projective plane that has one vertex part as its points, the other vertex part as its lines, such that a point and a line is connected if and only if they are incident in the projective plane. This leads to a -free graph with vertices and edges. Since this lower bound matches the upper bound given by I. Reiman, we have the asymptotic For , bipartite graphs with vertices on each side, edges, and no may again be constructed from finite geometry, by letting the vertices represent points and spheres (of a carefully chosen fixed radius) in a three-dimensional finite affine space, and letting the edges represent point-sphere incidences. More generally, consider and any . Let be the -element finite field, and be an element of multiplicative order , in the sense that form a -element subgroup of the