Local Conditions for Exponentially Many Subdivisions

Given a graph F , let st(F ) be the number of subdivisions of F , each with a different vertex set, which one can guarantee in a graph G in which every edge lies in at least t copies of F . In 1990, Tuza asked for which graphs F and large t, one has that st(F ) is exponential in a power of t. We show that, somewhat surprisingly, the only such F are complete graphs, and for every F which is not complete, st(F ) is polynomial in t. Further, for a natural strengthening of the local condition above, we also characterise those F for which st(F ) is exponential in a power of t. Mathematics Subject Classication. 05C10, 05C35, 05C83.