TITLE: Probabilistic method used to prove existence theorems
QUESTION [19 upvotes]: I am aiming for a "big list" of theorems using probability techniques to prove existence of some objects. And in each case, there is an interesting question -- can we find an explicit example? Was the probabilistic method the first one to come by or it was following the explicit construction? Are there any examples where it is proven that the example exists but its explicit construction is impossible? 
In wikipedia, on Probabilistic method, they do propose two examples due to Erdős:
FIRST ONE. For a complete graph on n vertices it is possible to color the edges of the graph in two colors so that there is no complete subgraph on r vertices which is monochromatic (every edge colored the same color).
Question: is there any explicit algorithm for such a coloring?
SECOND ONE.  The second problem also comes from graph theory and deals with a chromatic number of a graph: the minimal number of colors in which we can color the graph so that two adjacent vertices are colored differently. Given two positive integers $g$ and $k$, does there exist a graph containing only cycles of length at least $g$ such that its chromatic number of is at least $k$? It can be shown by the means of probabilistic method that such a graph exists for any values of $g$ and $k$.
Question: is there some algorithm to provide such a coloring?
I am not at all a specialist in graph theory, so I would like to hear the answers to thse questions as well as to see other examples which come from other branches of math.

REPLY [4 votes]: Somewhat surprisingly, Jean Bourgain's 1985 theorem about embedding finite metric spaces into Hilbert spaces was not mentioned yet. Since metrics have not mentioned at all in this thread, let me mention this example now. 
The topic of representing prescribed, possibly strange finite metric spaces 'as best one can' by non-strange, traditional metric spaces is important e.g. in computational biology. Very briefly, I think the essence of all of this is  

not to have to store a table of pairwise distances among any two of a large number of proteins, which after all would take space quadratic in the number of 'specimens', rather label/represent each protein by an element of a traditional metric space, and then calculate a distance the traditional way, from the traditional representatives, on an as-needed-basis. 

For this to result in only a small error of approximation (of the 'true' distance w.r.t. some new-fangled 'similarity distance' between proteins), one needs to know how well such 'representations'/'low-distortion embeddings' can be done in principle. 
Bourgain used the "probabilistic method" to prove a lower bound on how well this can be done.   
The publication is Jean Bourgain: On lipschitz embedding of finite metric spaces in Hilbert space. Israel Journal of Mathematics
March 1985, Volume 52, Issue 1–2, pp 46–52, and the abstract reads:

"It is shown than any $n$ point metric space is up to $\log n$ lipeomorphic to a subset of Hilbert space. We also exhibit an example of an $n$-point metric space which cannot be embedded in Hilbert space with distortion less than $(\log n)/(\log\log n)$, showing that the positive result is essentially best possible. The methods used are of probabilistic nature. For instance, to  construct our example, we make use of random graphs."

(emphasis added)