Proof of the Loebl–Komlós–Sós conjecture for large, dense graphs

The Loebl–Komlós–Sós conjecture states that for any integers kk and nn, if a graph GG on nn vertices has at least n/2n/2 vertices of degree at least kk, then GG contains as subgraphs all trees on k+1k+1 vertices. We prove this conjecture in the case when kk is linear in nn, and nn is sufficiently large.