most general way to generate pairwise independent random variables? Is there a sort of structure theorem for pairwise independent random variables or a very general way to create them?
I'm wondering because I find it difficult to come up with a lot of examples of nontrivial pairwise independent random variables. (by 'nontrivial', i mean not mutually independent)
one example (three r.v.):
X = face of dice 1
Y = face of dice 2
Z = X + Y mod 6
another example (three events) from some book:
Throw three coins. A = the number of heads is even, B = the first two flips are the same, C = the second two flips
are heads.
another example:
$A_{ij}$ = dice i and dice j having the same face
($A_{ij}$, $i\neq j$) form a set of pairwise independent events, but the triple ($A_{ij}$, $A_{jk}$, $A_{ki}$) is not mutually independent.
 A: I'm sure that Gil's answer is wise and that it is a good idea to look at Alon and Spencer's book.  Here also is a quick summary of what is going on.
Suppose that $X_1,\ldots,X_n$ are random variables, and suppose for simplicity that they take finitely many values.  Suppose that you prescribe the distribution of each $X_i$, and suppose also that you want the random variables to be pairwise independent or $k$-wise independent.  Then the constraints on the joint distribution are a finite list of equalities and inequalities.  The solution set is a polytope whose dimension is fairly predictable, and the fully independent distribution is always in the interior of this polytope.  If you are interested in $k$-wise independent distributions that are far from $k+1$-wise independent, then it can be difficult to determine what is achievable because the polytope is complicated.  (The vertices are a particularly interesting and non-trivial class:  $k$-wise independent distributions with small support.  These are called "weighted orthogonal arrays".)  However, if you're just intersted in examples, it is much easier to write down a small deviation of the fully independent distribution.  The deviation just satisfies linear equations.
For example, suppose that $X,Y,Z$ are three unbiased Bernoulli random variables (coin flips) that take values $0$ and $1$.  Then there are 8 probabilities $p_{ijk}$, one for each outcome $(X,Y,Z) = (i,j,k)$.  Then you can set
$$p_{ijk} = \frac18 + (-1)^{i+j+k}\epsilon. \qquad\qquad\qquad \text{(1)}$$
to get a pairwise independent but not independent distribution.  In this simple example, there is a 1-dimensional space of deviations and it is easy to compute how far you can vary the independent solution.  (Up to $|\epsilon| = \frac18$.)  In larger cases, the variations can be multidimensional and the polytope of deviations can be more complicated.
Addendum: If I have not made a mistake, all deviations for any finite list of discrete random variables are linear combinations of those of the form (1).  More precisely, given discrete random variables $X_1,\ldots,X_n$, let $f_i$ be some function of the value of $X_i$ which is 1 for one value, $-1$ for another value, and $0$ otherwise.  Then you can make deviations proportional to $\prod f_{i_j}$ as long as there are at least $k+1$ factors.  It looks like all deviations are a linear combination of those of this form.
A: One very useful construction: if $X_1,\ldots,X_n$ are i.i.d. RVs, uniform in $\{0,\ldots,q-1\}$ ($q$ prime), then two linear combinations $\sum a_i X_i$ and $\sum b_i X_i$ are independent iff the vectors $a$ and $b$ are linearly independent (all operations are modulo q).
If we take $q=2$, this means that using $n$ i.i.d unbiased coin flips we can generate $2^n-1$ pairwise independent RVs, by taking all nonzero linear combinations.
This can be further generalized and generally yields some connections between k-wise independence and error correcting codes.
A: Let $f$ be the density of $X$ on $\Omega$. Define $F_k: \Omega \times \Omega \to {\mathbb R}$ by $F_1(x, y) = f(x)$ and $F_2(x, y) = f(y)$. The two variables $X_1$ and $X_2$ with densities $F_1$ and $F_2$ should be independent by Fubini's theorem. That way you can produce $n$ copies of a given variable.
