Zoltan Lorant Nagy: Corollaries of a finite field contraction in the spirit of Mors and Furedi; and an alternative way to construct the projective planes PG(2,q)

We show an asymptotically tight result for the lower bound on  the number of copies of $K_{2,t}$ graphs in balanced bipartite graphs, in terms of the number of edges,   if the excess over the extremal number ex_bi( n+n, K_{2,t}) is at least of order n\sqrt{n}.

We also point out that the applied construction  yields a good bound on certain bipartite Ramsey numbers as well. 

Finally, we present a novel construction of the projective plane over a finite field via the solutions of a Vandermonde-type matrix-equality.