Cam anyone provide me the proof of:


  that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is connected.



Question: What is the cardinality of the family of unlabelled bipartite graphs on n vertices?
Answer: It seems that what Andrew wants to count are what are called in enumerative contexts
"bicolored graphs". A bicolored graph is a graph in which the vertices have been colored black and white so that every edge joins two vertices of different colors. A bipartite (or bicolorable) graph is a graph that has a bicoloring. A bicolorable graph with $k$ connected components has $2^k$ bicolorings. (In nonenumerative contexts the distinction between bipartite and bicolored is usually unimportant.) In addition, in counting  bicolored graphs one might or might not consider switching the two colors to give an equivalent graph. All of the versions of the enumeration problem have been solved. Counting unlabeled bicolored graphs (with no color-switching equivalence) is a straightforward application of Burnside's lemma; counting unlabeled bipartite graphs is tricky. 

It's not too hard to find appropriate references by searching MathSciNet. (Hint: "color" is sometimes spelled "colour".)

Incidentally, the number of labeled bicolored graphs on $n$ vertices is 
$$\sum_{m=0}^n 2^{m(n-m)}\binom{n}{m}.$$