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

TL;DR

This paper proves a structure theorem for graphs excluding a bounded degree tree, enabling the transfer of problems from sparse to dense graphs, and confirms the Erdős-Sós Conjecture for large bounded degree trees.

Contribution

It introduces a new structure theorem for T-free graphs that facilitates solving longstanding conjectures like Erdős-Sós for large bounded degree trees.

Findings

Proves a structure theorem for graphs with no bounded degree tree T

Transforms sparse T-free graph problems into dense graph problems

Provides a proof of the Erdős-Sós Conjecture for large, bounded degree trees

Abstract

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.