Ramsey型的单色子图及计数问题

Ramsey理论研究的是充分大的结构中给定子结构的存在性问题，有关Ramsey型问题的研究一直是极值组合中最活跃最重要的内容之一。本项目旨在研究与单色子图相关的一些Ramsey型问题。拟研究的主要内容：研究三角形对一些稠密图的Ramsey数的上界，进而说明对一些特殊图类Sidorenko猜想成立；研究Bollobás-Gyárfás猜想中单色k-连通图阶数的相关问题；研究单色子图的计数问题。这些问题的解决，不仅能推进Sidorenko猜想的解决，也能对单色子图的性质有更深入的刻画，从而丰富和完善图的Ramsey理论。

Ramsey theory studies the existence of a given substructure in a sufficiently large structure. The study of Ramsey-type problem has been one of the most active and important research fields in extremal combinatorics. The project aims at studying some Ramsey-type problems related to monochromatic subgraphs. The proposed research contents are as follows: provide an upper bound for triangle versus dense graphs, so as to verify the Sidorenko conjecture holds for some special graphs; study the order of the monochromatic k-connected subgraph concering in Bollobás-Gyárfás conjecture and the counting problems of monochromatic subgraph. As a result, the project can not only contribute to the earlier solution to Sidorenko conjecture, but also improve the the known results of monochromatic subgraph, thus generalize the Ramsey theory of graphs.