The edge-density for K_2,t minors

Let H be a graph. If G is an n-vertex simple graph that does not contain H as a minor, what is the maximum number of edges that G can have? This is at most linear in n, but the exact expression is known only for very few graphs H. For instance, when H is a complete graph Kt, the "natural" conjecture, (t-2)n- 1/2(t-1)-(t-2), is true only for t-7 and wildly false for large t, and this has rather dampened research in the area. Here we study the maximum number of edges when H is the complete bipartite graph K_2,t. We show that in this case, the analogous "natural" conjecture, 1/2(t+1)(n-1), is (for all t-2) the truth for infinitely many n.