The Erd\H os-Sós conjecture in dense graphs

Abstract: The Erd\H os--Sós conjecture states that every $n$-vertex graph with more than $(k-2)n/2$ edges contains every $k$-vertex tree. We prove that for every $γ$ there is an $n_0$ such that for all $n\ge n_0$ and $k \ge γn$ the conjecture holds. As a corollary of our result, we obtain a solution of a 51-year-old problem of Erd\H os and Graham on the multicolor Ramsey numbers of trees.