Topics in Extremal and Probabilistic Combinatorics via the study of uniform probability spaces with weak dependencies

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The aim of this project is to tackle different problems in Extremal and Probabilistic Combinatorics via the study of random combinatorial structures drawn from a non-standard probability space equipped with the uniform distribution. Typically, these probability measures present low or weak dependencies, resembling a product measure. This feature allows us to obtain both existential and enumerative results, as well as to determine the typical properties of random combinatorial objects. As an example, in the first part of the project we will study the existence of a spanning multicolored subgraph H with bounded maximum degree in an edge-colored graph G with large minimum degree. In this case, the probability space is given by the uniform measure on the set of copies of H in G. Advanced combinatorial techniques such as the Lu-Szekely approach to the Lopsided Lovasz Local Lemma are required to study this problem.