Coloring Graphs with Forbidden Structures and Investigations Related to Ramsey Theory

本研究计画旨在探讨结构图理论与极值组合学领域的各种基本问题。 这些问题通常与其他领域有关，包括理论计算机科学、几何学、信息论、调和分析和数论。这些问题的进展将促进我们对图论和组合学相关方面的理解。PI和她的合作者最近在其中大多数方面取得了进展。 预计对这些问题的进一步研究将导致新的方法和应用。研究生和本科生将在该奖项期间接受培训。该项目研究与著名的Hadwiger猜想，Erdös-Lovász Tihany猜想和Ramsey相关问题有关的问题。PI和她的合作者最近取得了自20世纪80年代以来的第一个渐近改进的色数没有团子的k阶使用新的技术。在这个计划中，PI和她的合作者计划探索以下问题：小阶图的线性Hadwiger猜想;通过研究Füredi-Gyárfás-Simonyi猜想，独立数为2的图的Hadwiger猜想的最小反例;特殊图族的Hadwiger猜想和Erdös-Lovász Tihany猜想;建立共临界图的最小边数的紧界;找到图，使得它的列表Ramsey数与它的经典Ramsey数相同。在这些问题上的进展可能会导致新的方法和途径的发展，这是太遥远的遥不可及的，并预计将允许进一步令人兴奋的发展。这一奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This research project aims to investigate a variety of fundamental problems in the areas of structural graph theory and extremal combinatorics. Often such problems are related to other areas including theoretical computer science, geometry, information theory, harmonic analysis and number theory. Progress on these problems will advance our understanding of related aspects of graph theory and combinatorics. The PI and her collaborators have recently made progress on most of them. It is expected that further work on these problems will lead to new methods and applications. Graduate and undergraduate students will be trained during this award.The project investigates problems related to the well-known Hadwiger's Conjecture, the Erdös-Lovász Tihany Conjecture and Ramsey related problems. The PI and her collaborators have recently made the first asymptotic improvement since 1980s on the chromatic number of graphs with no clique minor of order k using new techniques. In this proposal, the PI and her collaborators plan to explore the following problems: Linear Hadwiger’s Conjecture for graphs of small order; the minimal counterexamples to Hadwiger’s Conjecture for graphs with independence number two by working on the Füredi-Gyárfás-Simonyi Conjecture; Hadwiger’s Conjecture and the Erdös-Lovász Tihany Conjecture for special family of graphs; establishing tight bounds on the minimum number of edges of co-critical graphs; and finding graphs such that its list Ramsey number is the same as its classical Ramsey number. Progress on these problems will likely lead to the development of new methods and approaches that were too far out of reach before and are expected to allow further exciting developments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.