| Tóm tắt: | 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 characterize those F for which st (F) is exponential in a power of t.
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