Embedding spanning subgraphs into dense graphs via the regularity method

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The main theme of this research project is to determine essentially best-possible sufficient conditions for the embedding of large substructures into dense graphs. Building on the results and methods we obtained and developed for the proof of the Bollobäs-Komlös conjecture and problems in its vicinity during the first funding period of this project, we are now aiming for a variety of other central embedding problems in extremal combinatorics. Our key machinery will again be the regularity method, but it is clear that progress towards these goals will require substantial new ideas in order to cope with subgraphs of unbounded maximum degree and arbitrary order, sparse host graphs, median degree conditions or packings of exhaustive families of trees. In this way the overall aim of the project is to make progress on a set of important open problems and conjectures while simultaneously advancing the methodology.