拉姆齐理论和加莱-拉姆齐数及其应用研究

拉姆齐(Ramsey)理论由英国数学家和哲学家Frank P. Ramsey首先研究。由于内在的优美、广泛的应用和难以置信的困难，这个主题数十年来都是学术界一个研究热点。拉姆齐数及其相关参数在代数、集合、逻辑、遍历理论、信息理论、数论、集合论、有限域和理论计算科学方面有着重要应用。 .1967年，加莱(Gallai)首先研究了在传递有向条件下的无彩虹三角形这一特殊结构。加莱-拉姆齐数(Gallai-Ramsey)是经典拉姆齐数的一个自然和优美的推广，拉姆齐理论的彩虹推广有个动态综述。.在这个项目中，我们通过字典积染色研究加莱-拉姆齐数和Chung-Graham猜想，利用Lováz局部引理给出加莱-拉姆齐函数的渐进下界。我们也通过新的构造方法和反拉姆齐数研究Fujita-Magnant猜想。通过平衡染色，我们研究图运算和星图并；用正则引理和概率方法，研究给定最大度的的加莱-拉姆齐数。我们提出了图的边加莱-拉姆齐(Size Gallai-Ramsey)数，通过哈密顿和匹配工具研究加莱-拉姆齐o-序列，不交并的边加莱-拉姆齐数的极限，以及达到极限的图类。

Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. This subject has been a hot topic in mathematics for decades now due to their intrinsic beauty, wide applicability, and overwhelming difficulty despite somewhat misleadingly simple statements. From a theoretical perspective, Ramsey number and its related parameters have important applications in algebra, geometry, logic, ergodic theory, information theory, number theory, set theory and finite field. Ramsey number have applications on the fields of communications, information retrieval in computer science, decision-making, and theoretical computer science..A coloring of a graph is called rainbow if no two edges have the same color. Colorings of complete graphs that contain no rainbow triangle have very interesting and somewhat surprising structure. In 1967, Gallai first examined this structure under the guise of transitive orientations. Gallai -Ramsey number is a natural and nice generalization of the classical Ramsey number, and there is a dynamic survey of rainbow generalizations of Ramsey theory, including topics like Gallai-Ramsey numbers..In this research, we will study Gallai-Ramsey number and Chung-Graham conjecture by the lexicographic coloring, and give some asymptotic lower bounds for Gallai-Ramsey functions by Lováz Local Lemma. The Fujita-Magnant conjecture will be studied by some new construction method and anti-Ramsey number results. We also investigate the Gallai-Ramsey number for graph operations and union of star by balanced coloring, and the Gallai-Ramsey number with given maximum degree are studied by the Regularity Lemma and probabilistic method. We will introduce the concept of size Gallai-Ramsey number of graphs, and investigate Gallai-Ramsey o-sequence and some limits of this parameter of disjoint unions and the graphs attaining these limits by some Hamilton and matching tools.