Embedding Nearly Spanning Trees

The Erd{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with k edges as a subgraph. We prove that there are δ>0 and k₀∈ N such that the conjecture holds for every tree T with k ≥ k₀ edges and every graph G with |V(G)| ≤ (1+δ)|V(T)|.