Let G be a graph on n vertices. We call a subset A of the vertex set
V(G) \emph{k-small} if, for every vertex v∈A, deg(v)≤n−∣A∣+k. A subset B⊆V(G) is called \emph{k-large} if, for every vertex
u∈B, deg(u)≥∣B∣−k−1. Moreover, we denote by φk(G) the
minimum integer t such that there is a partition of V(G) into tk-small
sets, and by Ωk(G) the minimum integer t such that there is a
partition of V(G) into tk-large sets. In this paper, we will show tight
connections between k-small sets, respectively k-large sets, and the
k-independence number, the clique number and the chromatic number of a graph.
We shall develop greedy algorithms to compute in linear time both
φk(G) and Ωk(G) and prove various sharp inequalities
concerning these parameters, which we will use to obtain refinements of the
Caro-Wei Theorem, the Tur\'an Theorem and the Hansen-Zheng Theorem among other
things.Comment: 21 page