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

The Loebl–Komlós–Sós 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.