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.
Applying Lemma B.1 with W ′ = W, we have d(G − W ′) ≥ d/2, and by Corollary 2.4 with G = G − W ′, there exists a bipartite (ε1, ε2η)- expander H ⊆ G − W ′ with δ(H) ≥ d
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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.