On Dense Bipartite Graphs of Girth Eight and Upper Bounds for Certain Configurations in Planar Point–Line Systems

In earlier work we showed that ifG(m, n) is a bipartite graph with no 4-cycles or 6-cycles, and ifm<c1n2andn<c2m2, then the number of edgeseisO((mn)2/3). Here we give a more streamlined proof, obtaining some sharp results; for example, ifGhas minimum degree at least two thene⩽2 (mn)2/3, and this is a tight bound. Furthermore, one may allowO(mn) 6-cycles and still obtaine=O((mn)2/3). This leads us to conjecture that, ifG(m, n) is the point–line incidence graph of anynpoints andmlines in the plane, then the number of 6-cycles isO(mn). The main result of this paper is a proof that the number of 3-paths in such a graph isO(mn); this is related to the above conjecture.