In this talk I will present a preliminary work about fair division of indivisible goods. A standard approach in fair division problems is to consider a given fairness property (such as envy-freeness), and to look for an allocation that satisfies this property. In this work we investigate five such fairness properties (some of them being original or not very well-known) in a simple model of fair resource allocation of indivisible goods with additive preferences. We show how these properties are connected to each other, forming an ordered scale that can be used to characterize how conflicting the agents' preferences are: the less these preferences are conflicting, the more exigent property this instance will be able to satisfy, and the easier and more satisfactory the allocation will be. We also introduce some theoretical computational properties, and we further investigate a slightly richer model with k-additive preferences.