K_{s,t}-free graphs admit balanced clique subdivisions of order Ω(d^{s/(2(s-1))}) and satisfy a (s/(2(s-1)) - o(1)) log d lower bound on the sum of reciprocal cycle lengths.
Thomassen, Problems 20 and 21, in Graphs, Hypergraphs and Applications
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Balanced clique subdivisions and cycles lengths in $K_{s, t}$-free graphs
K_{s,t}-free graphs admit balanced clique subdivisions of order Ω(d^{s/(2(s-1))}) and satisfy a (s/(2(s-1)) - o(1)) log d lower bound on the sum of reciprocal cycle lengths.