An approximate version of the Loebl–Komlós–Sós conjecture

Loebl, Komlós, and Sós conjectured that if at least half of the vertices of a graph G have degree at least some k∈N, then every tree with at most k edges is a subgraph of G. Our main result is an approximate version of this conjecture for large enough n=|V(G)|, assumed that n=O(k).Our result implies an asymptotic bound for the Ramsey number of trees. We prove that r(Tk,Tm)⩽k+m+o(k+m), as k+m→∞.