This BCs thesis deals with topics from graph theory. Ramsey theory in its most basic form deals with the problem of determining the minimal positive integer, such that for any edge-coloring of the complete graph of this size with a prescribed number of colors one can find a subgraph of predefined size all of whose edges are of the same colour. These minimal sizes are called Ramsey numbers.
In this BCs thesis we present basic notions of graph theory needed to understand the basic theorem of Ramsey theory and its proof. We discuss the problem of determining bounds for Ramsey numbers. We present Ramsey numbers found till today, and for some of them we prove that they are correct. We indicate some possibilities of how to use Ramsey theory with a few examples.