The Gyárfás-Sumner conjecture says that for every forest H and every integer k, if G is H-free and does not contain a clique on k vertices then it has bounded chromatic number. (A graph is H-free if it does not contain an induced copy of H.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest H, if G is H-free and does not contain $K_{t,t}$ as a subgraph then it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened Rödl's result, showing that for every forest H, the bound on chromatic number can be taken to be polynomial in t. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree H of radius two and integer $d\ge 2$, if G is H-free and does not contain as a subgraph the complete d-partite graph with parts of cardinality t, then its chromatic number is at most polynomial in t.

Gyárfás-Sumner 猜想表明，对于每个森林H和每个整数k，如果G是H自由的并且在k 个顶点上不包含团，那么它具有有界色数。（如果一个图不包含H的导出副本，则该图是H 自由的。）Kierstead 和 Penrice 针对半径最多为 2 的树证明了这一点，但除此之外，该猜想仅适用于几种简单类型的森林。如果我们排除一个完整的二分子图而不是一个集团，我们就会知道更多：Rödl 表明，对于每个森林H，如果G是H自由的并且不包含$K_{t,t}$作为子图，那么它具有有界色数。在 Sophie Spirkl 的早期论文中，我们强化了 Rödl 的结果，表明对于每个森林H，色数上的界限可以视为t中的多项式。在本文中，我们证明了 Kierstead-Penrice 定理的相关强化，表明对于半径为 2 且整数的每棵树H$d\ge 2$，如果G是无H的并且不包含作为子图的具有部分基数t的完整d分图，则其色数至多是t中的多项式。