The extremal function for complete minors
Journal of Combinatorial Theory Series B
Minors in graphs of large girth
Random Structures & Algorithms
A High Girth Graph Construction
SIAM Journal on Discrete Mathematics
Forcing unbalanced complete bipartite minors
European Journal of Combinatorics
Light subgraphs of order at most 3 in large maps of minimum degree 5 on compact 2-manifolds
European Journal of Combinatorics - Special issue: Topological graph theory II
Improved Bounds for Topological Cliques in Graphs of Large Girth
SIAM Journal on Discrete Mathematics
The Extremal Function For Noncomplete Minors
Combinatorica
Compact topological minors in graphs
Journal of Graph Theory
Hitting diamonds and growing cacti
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Hitting and harvesting pumpkins
ESA'11 Proceedings of the 19th European conference on Algorithms
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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)@?logn vertices. The logarithmic dependence on n is best possible (for fixed t). In general, we prove that f(t)@?2^t^-^1+@e. For t@?4, we determine the least value of f(t); in particular, f(3)=2+@e and f(4)=4+@e. 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).