The approximate Loebl-Komlós-Sós conjecture III: The finer structure of LKS graphs

This is the third of a series of four papers in which we prove the following relaxation ofthe Loebl–Komlós–S ́os Conjecture: For everyα >0 there exists a numberk0such that foreveryk > k0everyn-vertex graphGwith at least (12+α)nvertices of degree at least (1 +α)kcontains each treeTof orderkas a subgraph.In the first paper of the series, we gave a decomposition of the graphGinto several partsof different characteristics. In the second paper, we found a combinatorial structure inside thedecomposition. In this paper, we will give a refinement of this structure. In the forthcomingfourth paper, the refined structure will be used for embedding the treeT.