  Consider the following information theoretic setup wherein independent
codebooks of N correlated random variables are generated according to their
respective marginals. The problem of determining the conditions on the rates of
codebooks to ensure the existence of at least one codeword tuple which is
jointly typical with respect to a given joint density (called the multivariate
covering lemma) has been studied fairly well and the associated rate regions
have found applications in several source coding scenarios. However, several
multiterminal source coding applications, such as the general multi-user
Gray-Wyner network, require joint typicality only within subsets of codewords
transmitted. Motivated by such applications, we ask ourselves the conditions on
the rates to ensure the existence of at least one codeword tuple which is
jointly typical within subsets according to given per subset joint densities.
This report focuses primarily on deriving a new achievable rate region for this
problem which strictly improves upon the direct extension of the multivariate
covering lemma, which has quite popularly been used in several earlier work.
Towards proving this result, we derive two important results called `subset
typicality lemmas' which can potentially have broader applicability in more
general scenarios beyond what is considered in this report. We finally apply
the results therein to derive a new achievable region for the general
multi-user Gray-Wyner network.
