Small minors in dense graphs

A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe functions f and h such that every graph with n vertices and average degree at least f ( t ) contains a K t -model with at most h ( t ) ⋅ log n vertices. The logarithmic dependence on n is best possible (for fixed t ). In general, we prove that f ( t ) ≤ 2 t − 1 + ε . For t ≤ 4 , we determine the least value of f ( t ) ; in particular, f ( 3 ) = 2 + ε and f ( 4 ) = 4 + ε . For t ≤ 4 , we establish similar results for graphs embedded on surfaces, where the size of the K t -model is bounded (for fixed t ).