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 K-t. 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,K-1. 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.
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