For {P_s,K_{t,t}}-free graphs, the maximum path length is at most 2^{ω(G)^c}, and treedepth is clique-polynomial.
Minor-matching hypertree width
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Excluding paths and bicliques
For {P_s,K_{t,t}}-free graphs, the maximum path length is at most 2^{ω(G)^c}, and treedepth is clique-polynomial.