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Let H be a tree. It was proved by Rodl that graphs that do not contain H asan induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$as a subgraph, have bounded chromatic number. Kierstead and Penricestrengthened this, showing that such graphs have bounded degeneracy. Here wegive a further strengthening, proving that for every tree H, the degeneracy isat most polynomial in t. This answers a question of Bonamy, Pilipczuk,Rzazewski, Thomasse and Walczak.

Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree