Lecture 6 : Forbidden Bipartite Cycles and Cliques

To get a matching lower bound, we showed that as long as we can construct a finite projective plane of order d, then we have a bipartite graph that is C4-free and has Θ(n ) edges where the number of vertices n = d + d + 1. In this lecture we will construct such objects when d is a prime power. This construction is also due to Reiman 1958. Let q be a prime power, i.e. q = p where p is a prime and a ∈ N. Then there exists a finite field Fq with q elements. (If q is a prime, then we can simply take the set of integers modulo q. Otherwise to construct Fq requires a little bit of algebra. We won’t do it here but it can be found in almost any algebra textbooks.) In fact, for what we are going to do below, it is enough to simply think q as a prime number. We build a projective plane of order q as follows. Let V = Fq, the vector space of size q over the field Fq. So an element x ∈ V has the form of (a, b, c) where a, b, c ∈ Fq. Recall that a linear subspace S is defined by the following three properties.