Calibers, free sequences and density

We prove, among others, the following theorems: (I) s a T3 space with no free sequence of length λ and (λ,λ,κ) is a caliber of X then d(X)⩽μ<λ for some cardinal μ<κ. s T3 and X=⋃C with |C|<κ and C is compact with no free sequences of length μ for each C∈C, moreover (μ,κ) is a caliber of X then d(X)<κ. s T3 and X=⋃C with |C|⩽κ and C is compact with no free sequences of length κ for every C∈C, moreover κ is a caliber of X then d(X)<κ. results provide strenthenings and generalizations of some results of Šapirovskiı̆ [Uč. Zap. Riga Univ. 3 (1976) 88–89] and of Arhangelskiı̆ [Topology Appl. 104 (2000) 13–16], respectively.