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.