Chernoff-type inequalities are used to show that the probability that a sum of independent random variables deviates significantly from its expected value is exponentially small in the expected value and the deviation. Is there a Chernoff-type inequality for any sum of pairwise independent random variables? In other words, is there a result that shows the following: the probability that a sum of pairwise independent random variables deviates from its expected value is exponentially small in the expected value and the deviation?
Pairwise independence is not enough for a Chernoff-type bound on the expectation.
This follows from the fact that there are $poly(n)$-size sample spaces on $n$ 0-1 random variables, where all the variables are pairwise independent, and each is variable is uniform (it is $1$ with probability $1/2$). So the expected value of their sum is $n/2$. But because there are only $poly(n)$ possible events in the sample space, even the probability that a sum is exactly a particular value $v$ is at least $1/poly(n)$ (hence, it can't be at most $1/exp(n)$).
For a reference to this sample space construction, see pages 11-12 in this survey.
If you have pairwise independence, then you can bound the variance of the sum, and thus get a concentration bound using Chebyshev's inequality.
There are all kinds of results of this kind in the Dubhashi-Panconesi book. One standard reference of this kind is the 1993 work by Schmidt, Siegel and Srinivasan titled (appropriately enough) "Chernoff-Hoeffding bounds for applications with limited independence"