A local approach to the Erd\H{o}s-S\'os conjecture

A famous conjecture of Erd\H{o}s and S'os states that every graph with average degree more than $k-1$ contains all trees with $k$ edges as subgraphs. We prove that the Erd\H{o}s-S'os conjecture holds approximately, if the size of the embedded tree is linear in the size of the graph, and the maximum degree of the tree is sublinear.