Let M be a matroid and let ρ be its rank function, Ingleton's inequality states that for any subsets X1, X2, X3 and X4 in the support of M, the inequality Aubrey William Ingleton, an English mathematician, wrote an important paper in 1969[1] in which he surveyed the representability problem in matroids.
Although the article is mainly expository, in this paper Ingleton stated and proved Ingleton's inequality, which has found interesting applications in information theory, matroid theory, and network coding.
Perhaps, the more interesting application of Ingleton's inequality concerns the computation of network coding capacities.
Linear coding solutions are constrained by the inequality and it has an important consequence: For definitions see, e.g.[6] Theorem (Ingleton's inequality):[7] Let M be a representable matroid with rank function ρ and let X1, X2, X3 and X4 be subsets of the support set of M, denoted by the symbol E(M).
Finally, if we define Vi = {vr : r ∈ Xi } for i = 1,2,3,4, then by last inequality and the item (4) of the above proposition, we get the result.