# What's the difference between Ramsey theory and Extremal graph theory?

Wikipedia teaches us that problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?"

It also teaches us that "Extremal graph theory studies extremal (maximal or minimal) graphs which satisfy a certain property". This sounds very similar. Let's take Turan's theorem (the basic example of an EGT theorem) - it gives a bound on the number of edges in a clique-free graph (for a given clique size). Put differently, it tells us how many elements (edges) of a certain structure (graph with fixed number of vertices) must be there to guarantee that a particular property (existence of a clique of a specific size) will hold.

One can argue that in EGT we try to understand the structure of the extremal objects, but I don't see why Ramsey theory should not try to do the same thing, and also I don't see how this is done in the (beautiful) Erdős–Stone theorem which is considered a fundamental theorem of EGT.

So is there a real difference or is it just a matter of taste?

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I think EGT is a subset of Ramsey theory. In Ramsey theory, we are concerned with structures other then graphs as well, for example in linear equations (Rado's theorem). – Shahab Sep 1 '11 at 11:28

Extremal graph theory tends to focus on graph-specific questions of minimality/maximality. Questions like how do I guarantee that every vertex in a subgraph has a given minimum degree? How many disjoint cycles must a graph of size $m$ have? Or given a $k$-regular graph what is its minimum/maximum girth? These are questions dealing with the explicit constructions and natures of the objects of graph theory. Sometimes the results that are obtained find applicability in "real" world applications, but often they simply help us to better understand the extremal nature of regular graphs.