Ramsey-type problems in random and in randomly perturbed discrete structures, and their sharp thresholds

研究随机离散结构及其与确定性对应结构的相互作用是极值组合学和概率组合学的一个活跃研究方向。几位研究人员观察到，将确定性对象与随机结构相结合通常会产生更好的结果。Bohman， Frieze和Martin在2001年引入了随机摄动图模型，并证明了在一个小线性度的图中随机添加线性多条边很可能得到哈密顿图。近年来，随机摄动模型的研究已经扩展到超图和整数集，是一个活跃的研究方向，有许多有趣的见解。本项目的目的是进一步推进这一领域，特别关注随机摄动离散结构（图、超图、整数集）周围的ramsey型问题及其与纯随机模型中相应语句的关系。此外，应该研究所谓的急剧阈值（结构经历其特征属性的急剧变化）现象，ramsey型问题似乎是一个具有挑战性的试验台。

The study of random discrete structures and their interplay with deterministic counterparts is an active stream of research within extremal and probabilistic combinatorics. It has been observed by several researchers that the combination of a deterministic object with a random structure often leads to much better results. Bohman, Frieze and Martin introduced in 2001 the randomly perturbed graph model and proved that adding to a graph of small linear degree linearly many edges at random results very likely in a Hamiltonian graph. In the recent few years the study of randomly perturbed models has been extended to hypergraphs and sets of integers and is now an active research direction with many interesting insights. The purpose of this project is to further advance this area, especially focusing on Ramsey-type problems around randomly perturbed discrete structures (graphs, hypergraphs, sets of integers) and their relation to the corresponding statements in purely random models. Moreover, the phenomenon of the so-called sharp thresholds (where the structure undergoes a dramatic change of its characteristic properties) should be studied and Ramsey-type problems appear to be a challenging testbed.