There exists an absolute constant g0 such that every graph with minimum degree at least k and girth at least g0 contains an induced subdivision of K_{k+1}.
Wang, Balanced subdivisions of a large clique in graphs with high average degree, SIAM J
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Induced subdivisions in graphs of large girth
There exists an absolute constant g0 such that every graph with minimum degree at least k and girth at least g0 contains an induced subdivision of K_{k+1}.