Every K_{s,t}-free graph of average degree Ω(h^{2(s-1)} log^{7(s-1)} h) and every C_{2k}-free graph of average degree Ω(h log^5 h) contains an induced subdivision of K_h, both nearly optimal.
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Nearly tight bounds for induced subdivisions
Every K_{s,t}-free graph of average degree Ω(h^{2(s-1)} log^{7(s-1)} h) and every C_{2k}-free graph of average degree Ω(h log^5 h) contains an induced subdivision of K_h, both nearly optimal.