Turán数、蕴含数和蕴含-Ramsey数的研究

研究极值图论与图的度序列中当今国际同行关注的几个活跃问题：完全确定k个相同星图S的并图的Turán数；确定任意星森林F的Turán数的新下界，进而探索完全确定任意星森林F的Turán数；改进任意图H的蕴含数的上界使得ω(n)为仅依赖于|H|的一个常数；对充分大的n，确定任意图H的蕴含数和“包含所有k阶l-树作为子图”的图性质的蕴含数；对较小的m和n，确定Ks,t的二部蕴含数，以及完全确定“是群A-连通”的图性质的二部蕴含数；探索一般图G1和G2的蕴含-Ramsey数的新下界；对重要的图类G1和G2，确定G1和G2的蕴含-Ramsey数等，以推进图的Turán数、蕴含数和蕴含-Ramsey数的研究和发展，将对图论和理论计算机科学具有重要理论意义和应用前景。

We will investigate the following several active problems in extremal graph theory and degree sequences of graphs which are concerned (or interested) by international counterparts at present: completely determine the Turán number of the disjoint union of k copies of the star S; determine the new lower bound of the Turán number of any star forest F, and further investigate to completely determine the Turán number of any star forest F; improve the upper bound of the potential number of any graph H so thatω(n) only depends on |H|; determine the potential numbers of any graph H and of the property “containing all l-trees of order k as subgraphs” for n sufficiently large; determine the bipartite-potential number of Ks,t for smaller m and n, completely determine the bipartite-potential number of the property “being group A-connected”; investigate the new lower bound of the potential-Ramsey number of general graphs G1 and G2; determine the potential-Ramsey number of G1 and G2 for important graphs G1 and G2, etc. These results will improve the research and development of the Turán number、the potential number and the potential-Ramsey number of graphs, and also will be important to graph theory and theoretical computer science in theoretical significances and application prospects.