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

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