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

该项目的目的是通过研究从配备均匀分布的非标准概率空间中提取的随机组合结构来解决极值和概率组合学中的不同问题。通常，这些概率度量呈现低或弱的依赖性，类似于乘积度量。这一特点使我们能够同时获得存在性和枚举性的结果，以及确定随机组合对象的典型性质。作为一个例子，在第一部分的项目中，我们将研究的存在性的生成多色子图H的最大度有界的边着色图G具有大的最小度。在这种情况下，概率空间由H在G中的拷贝集上的一致测度给出。需要先进的组合技术，如Lu-Szekely方法的Lopsided Lovasz局部引理来研究这个问题。

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.