Abstract: Ramsey Theory reasures us that the complete disorder is impossible. In graph theory setting this means that for a given a graph G and an integer k>1, in any coloring of the edges of the complete graph K_N by k colors, there exists a monochromatic copy of G provided N is large. The smallest integer N with this property is called the Ramsey number r_k(G).

In the first half of my talk, we will briefly survey the most interesting results when k=2. In the second half, we look at the case when k>2. Here we do not know much even if G is a very simple graph, e.g., a cycle.