ABSTRACT
This chapter discusses many of Erdos favorite problems in extremal graph theory. There is considerable overlap between extremal problems and the Ramsey problems. Extremal graph theory deals with the inevitable occurrence of some specified structure when the edge density in a graph exceeds certain threshold. On the other hand, Ramsey theory is concerned with the inevitable occurrence of certain substructures in any finite partition of a large structure. Erdos often compared his failure to realize that the C4 -free result was just the tip of the vast expanse of extremal graph theory to the corresponding failure of Sir William Crookes to realize the importance of X-rays when he first discovered them. In Turan’s classical result for Kk -free graphs, the extremal graph which achieves the maximum number of edges and contains no Kk has a large independent set.
