The approximate Loebl-Koml\'os-S\'os Conjecture IV: Embedding techniques   and the proof of the main result
  This is the last paper of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number~$k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph.   In the first two papers of this series, we decomposed the host graph $G$, and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure, and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlos-Sos Conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations. 
Jan Hladk\'y, J\'anos Koml\'os, Diana Piguet, Mikl\'os Simonovits,   Maya J. Stein, Endre Szemer\'edi
math.CO