Title: Problems and results in combinatorial analysis and graph theory
Abstract: Publisher Summary This chapter discusses the problems and results that arise in combinatorial analysis and the graph theory, illustrating two recent problems. The first problem assumes G be a graph each vertex of which has degree not exceeding n . It is true that if G has more than 5 n 2 /4 edges, then G contains two strongly independent edges. This means that two edges, which are vertex disjoint and for which the subgraph of G induced by the vertices of these two edges contains only these two edges. This, as well as the second problem, naturally leads to the problem of Ramsey numbers. The chapter states some of these problems and gives out a list of references. Referring to the academic failures in terms of proving simple inequalities between Ramsey numbers, the chapter states that the main difficulty is perhaps the lack of constructive methods.