Given two graphs G and H, the Ramsey number R(G,H) is defined as the minimum number of vertices n such that every red-blue edge-coloring of the complete graph K_n contains either a red copy of G or a blue copy of H. If G is isomorphic to H, then we write R(G) for short. For any positive integer k, the Gallai-Ramsey number gr_k(G:H) is the minimum number of vertices n such that any exact k-edge coloring of K_n contains either a rainbow copy of G or a monochromatic copy of H. The bipartite Gallai-Ramsey number bgr_k(G:H) is the minimum number of vertices n such that for every N greater than or equal to n, any exact k-edge coloring of the complete bipartite graph K_{N,N} contains a rainbow copy of G or a monochromatic copy of H. In this talk, we describe the structures of a complete bipartite graph K_{n,n} without small rainbow subgraphs and determine the Ramsey and bipartite Gallai-Ramsey numbers for some families of graphs.

魏美芹，上海海事大学理学院讲师，博士毕业于南开大学组合数学中心，师从李学良教授，2023-2024学年于华东师范大学数学科学学院国内访学，师从吕长虹教授。主要从事图的拉姆齐数和加莱-拉姆齐数以及图能量极值问题的研究，主持国家自然科学基金青年基金项目一项，获得上海市启明星（扬帆专项），发表SCI论文十余篇。