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

本课题的目的是通过研究均匀分布的非标准概率空间中的随机组合结构来解决极值组合学和概率组合学中的不同问题。通常，这些概率度量表现出较低或较弱的依赖性，类似于产品度量。这个特性允许我们获得存在和枚举结果，以及确定随机组合对象的典型属性。作为一个例子，在项目的第一部分，我们将研究最大有界度的生成多色子图H在最小度较大的边色图G中的存在性。在这种情况下，概率空间是由H在g中的副本集合上的一致测度给出的。研究这个问题需要先进的组合技术，如不平衡Lovasz局部引理的Lu-Szekely方法。

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.