Intersection theorems for systems of sets (III)

A system or family ( A γ : γ∈ N ) of sets A γ , indexed by the elements of a set N , is called an ( a, b)-system if ¦N¦ = a and ¦A γ ¦ = b for γ ∈ N . Expressions such as “( a, )-system” are self-explanatory. The system (A γ : γ∈ N ) is called a δ-system [1] if A μ ∩ A γ = A p ∩ A σ whenever μ, γ, ρ, σ ∈ N ; μ≠ γ; ρ ≠ σ. If we want to indicate the cardinality ¦N¦ of the index set N then we speak of a δ( ¦N¦ ) system. In [1] conditions on cardinals a, b, c were obtained which imply that every (a, b)-system contains a δ(c)-subsystem. In [2], for every choice of cardinals b, c such that the least cardinal a = f δ (b, c) was determined which has the property that every (a,