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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 contains all trees with 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