The edge-density for K2, 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− 12(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 K2,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.