An embedding of a graph H in a graph G is an injective mapping of the vertices of H to the vertices of G such that edges of H are mapped to edges of G. Embedding problems have been extensively studied. A very powerful tool in this area is Szemeredi's Regularity Temma. It approximates the host graph G by a quasirandom graph which inherits many of the properties of G. Unfortunately the direct use of Szemeredi's Regularity Lemma is useless if the host graph G is sparse.
During the talk I shall expose a technique to deal with embedding trees in sparse graphs. This technique has been developed by Ajtai, Komlos,Simonovits and Szemeredi to solve the Erdos-Sos conjecture. Presently the author together with Hladky, Komlos, Simonovits, Stein and Szemeredi apply this method to solve the related conjecture of Loebl, Komlos and Sos (approximate version).