Polynomial bounds for chromatic number VIII. Excluding a path and a complete multipartite graph - Nguyen, T; Scott, AD; Seymour, P (Wiley, 2024)
We prove that for every path H, and every integer d, there is a polynomial f such that every graph G with chromatic number greater than f(t) either contains H as an induced subgraph, or contains as a subgraph the complete d-partite graph with parts of cardinality t. For t = 1 and general d this is a classical theorem of Gyárfás, and for d = 2 and general t this is a theorem of Bonamy et al.