In this paper we provide three new characterizations of largeness of the core. The first characterization of largeness of the core is based on minimal covers of the grand coalition and associated inequalities. The second characterization shows the relation between the bases that provide core elements of the game and the bases that provide core elements of the games that are obtained from the original one by increasing the value of the grand coalition. The third characterization of largeness of the core is based on the idea that if a base of the grand coalition does not provide a core element of the game, it should not provide a core element of a game which differs from the original one only by an increase of the value of the grand coalition. Based on these new characterizations, we show the equivalence between largeness of the core and stability of the core for games with at most 5 players and we revise some results on the literature concerning largeness of the core.