Excluding paths and bicliques

Classes of graphs excluding a path and a biclique as induced subgraphs are extensively studied in the literature. One of the key structural results for such graphs is a Ramsey-type result due to Galvin, Rival, and Sands (1982), establishing the existence of a function $f$ bounding the maximum length of a path in terms of clique number $\omega$. We improve the best known bound on $f$ to a function that is a singly exponential in $\omega^c$, for some constant $c$, which we show is best possible, up to optimizing $c$.
Our approach also has consequences for treedepth. In particular, we show that, for graphs excluding a path and a biclique as induced subgraphs, treedepth is bounded by a polynomial function of clique number. In turn, this result implies that every hereditary graph class that admits a function bounding treedepth of graphs in the class in terms of clique number, admits a polynomial such function. This gives a treedepth analogue of a recent result on pathwidth due to Hajebi (2025).