By an r-graph, we mean a finite set V of elements called vertices and a collection of some of the r-subsets of V called edges with the property that each vertex is incident with at least one edge. An A-chromatic r-graph is an r-graph all of whose edges are coloured A.

Theorem. Let G1, …, Gt denote r-graphs. There exists a nonempty class of r-graphs such that for each if the edges of G are painted arbitrarily in t colours A1, …, At, then for at least one i in {1, …, t}, G has an Ai-chromatic r-subgraph which is isomorphic to Gi.

Of special interest in summability theory are those conservative matrices possessing the "mean-value property". If cA={x: Ax ∊ c} denotes the convergence domain of a conservative matrix A, then A has the mean-value property in case, for each x in cA, there exists M = M(A, x) > 0 such that

1

This property has been considered by many writers and has been shown, among other things, to be equivalent to the requirement that the matrix be associative, i.e., for each x in cA

2

R is a ring and M is a right R-module for which Rl = {m ∊ M | mR = 0} is the zero submodule. Let and be the injective hull and the quasi-injective hull of M respectively. Then where [1]. The ring plays an important role, in many cases, in the studying of R especially when D is a division ring. For x ∊ M, we denote the annihilator of x in R by xγ = {r ∊ R | xr=0}, whereas xγl={m ∊ M | mxγ=0}. If N is a submodule of M and x ∊ M, x-1(N) is the right ideal in R consisting of elements r in R where xr ∊ N.

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function

1

where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.