Coloring Graphs with Forbidden Structures and Investigations Related to Ramsey Theory

该研究项目旨在研究结构图论和极值组合学领域中的各种基本问题。这类问题通常涉及其他领域，包括理论计算机科学、几何、信息论、调和分析和数论。在这些问题上的进展将促进我们对图论和组合学的相关方面的理解。PI和她的合作者最近在其中大多数方面取得了进展。预计在这些问题上的进一步工作将导致新的方法和应用。研究生和本科生将在这一奖项中接受培训。该项目调查与著名的哈德威格猜想、埃德·S-洛瓦兹·蒂哈尼猜想和拉姆齐相关问题有关的问题。最近，Pi和她的合作者利用新的技术首次对图的色数进行了自20世纪80年代以来的渐近改进。在这个建议中，Pi和她的合作者计划探索以下问题：对于小阶图的线性Hadwiger猜想；通过Füredi-GyárfáS-Simonyi猜想对独立数为2的图的Hadwiger猜想的最小反例；对于特殊的图族，Hadwiger猜想和ErdöS-Lovász Tihany猜想；建立关于上临界图的最小边数的紧界；以及找到其列表Ramsey数与其经典Ramsey数相同的图。在这些问题上的进展可能会导致开发以前遥不可及的新方法和途径，并有望允许进一步令人兴奋的发展。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.