Let R(s,t) denote the minimum number of vertices a complete graph must have to guarantee the existence of a red K_s or a blue K_t, if one colors all of the edges of K_n red or blue. It turns out to be very hard to compute the exact values of R(s,t). In fact, we only know a few values of R(s,t) exactly. We study the behavior of R(n,n), the central Ramsey numbers, as n tends to infinity, instead.

We know that 2^(n/2)< R(n,n) < 4^n, but not much have been improved on these bounds in the last half century.

One can also study Ramsey theory for more colors. For instance, it is known that R(3,3,3) = 17. It is also interesting to study what R(s,s,….s) is, asymptotically.

Size Ramsey numbers: The size Ramsey number of a graph H is the minimum number of edges a graph must have to be guaranteed an H as a subgraph.

Ramsey numbers for hypergraphs. The number R_k(s,n) is the minimum number N such that every red blue coloring of an N element set contain either a red set of size s or a blue set of size n, where a set is called red (blue) if all of the k-tuples in this set are red (blue). Erdos, Hajnal, and Rado showed that there are positive constants c and c’ such that

2^(cn^2) < r_3(n,n) < 2^2^(c’n)

We want to improve these bounds