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In a series of four papers we prove the following relaxation of theLoebl-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 eachtree $T$ of order $k$ as a subgraph.  The method to prove our result follows a strategy similar to approaches thatemploy the Szemer\'edi regularity lemma: we decompose the graph $G$, find asuitable combinatorial structure inside the decomposition, and then embed thetree $T$ into $G$ using this structure. Since for sparse graphs $G$, thedecomposition given by the regularity lemma is not helpful, we use a moregeneral decomposition technique. We show that each graph can be decomposed intovertices of huge degree, regular pairs (in the sense of the regularity lemma),and two other objects each exhibiting certain expansion properties. In thispaper, we introduce this novel decomposition technique. In the three follow-uppapers, we find a combinatorial structure suitable inside the decomposition,which we then use for embedding the tree.

The approximate Loebl-Koml\'os-S\'os Conjecture I: The sparse decomposition