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The present paper offers, in its first part, a unified approach for thederivation of families of inequalities for set functions which satisfysub/supermodularity properties. It applies this approach for the derivation ofinformation inequalities with Shannon information measures. Connections of theconsidered approach to a generalized version of Shearer's lemma, and otherrelated results in the literature are considered. Some of the derivedinformation inequalities are new, and also known results (such as a generalizedversion of Han's inequality) are reproduced in a simple and unified way. In itssecond part, this paper applies the generalized Han's inequality to analyze aproblem in extremal graph theory. This problem is motivated and analyzed fromthe perspective of information theory, and the analysis leads to generalizedand refined bounds. The two parts of this paper are meant to be independentlyaccessible to the reader.

Information Inequalities via Submodularity and a Problem in Extremal Graph Theory