Hyperstability in the Erd\H{o}s-S\'os Conjecture

A rough structure theorem is proved for graphs $G$ containing no copy of a bounded degree tree $T$: from any such $G$, one can delete $o(|G||T|)$ edges in order to get a subgraph all of whose connected components have a cover of order $3|T|$. This theorem has the ability to turn questions about sparse $T$-free graphs (about which relatively little is known), into questions about dense $T$-free graphs (for which we have powerful techniques like regularity). There are various applications, the most notable being a proof of the Erd\H{o}s-S\'os Conjecture for large, bounded degree trees.