This paper addresses the following question for a given graph H: What is the minimum number f(H) such that every graph with average degree at least f(H) contains H as a minor? Due to connections with Hadwiger's conjecture, this question has been studied in depth when H is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined f(H) when H has a super-linear number of edges. We focus on the case when H has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if H has t vertices and average degree d at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case when H has small average degree, we prove that if H has t vertices and q edges, then f(H) ⩽ t + 6.291q (where the coefficient of 1 in the t term is best possible).

A classical result of Robertson and Seymour states that the set of graphs containing a fixed planar graph H as a minor has the so-called Erdős–Pósa property; namely, there exists a function f depending only on H such that, for every graph G and every positive integer k, the graph G has k vertex-disjoint subgraphs each containing H as a minor, or there exists a subset X of vertices of G with |X| ≤ f(k) such that G − X has no H-minor (see Robertson and Seymour, J. Combin. Theory Ser. B41 (1986) 92–114). While the best function f currently known is exponential in k, a O(k log k) bound is known in the special case where H is a forest. This is a consequence of a theorem of Bienstock, Robertson, Seymour and Thomas on the pathwidth of graphs with an excluded forest-minor. In this paper we show that the function f can be taken to be linear when H is a forest. This is best possible in the sense that no linear bound is possible if H has a cycle.