We construct an incidence structure using certain points and lines in finite projective spaces. The structural properties of the associated bipartite incidence graphs are analyzed. These n × n bipartite graphs provide constructions of C6-free graphs with Ω(n 4/3) edges, C10-free graphs with Ω(n 6/5) edges, and Θ(7, 7, 7)-free graphs with Ω(n 8/7) edges. Each of these bounds is sharp in order of magnitude.

We consider extensions of Turán’s original theorem of 1941 to planar grids. For a complete k x m array of vertices, we establish in Proposition 4.3 an exact formula for the maximal number of edges possible without any square regions. We establish with Theorem 4.12 an upper bound and with Theorem 4.15 an asymptotic lower bound for the maximal number of edges on a general grid graph with n vertices and no rectangles.