Conflict Free Colorings of (strongly) Almost Disjoint Set-systems

f : ∪A → ρ is called a conflict free coloring of the set-system A (with ρ colors) if ∀A ∈ A ∃ ζ < ρ ( |A ∩ f−1{ζ}| = 1 ). The conflict free chromatic number χCF(A) of A is the smallest ρ for which A admits a conflict free coloring with ρ colors. A is a (λ, κ, μ)-system if |A| = λ, |A| = κ for all A ∈ A, and A is μ-almost disjoint, i.e. |A ∩ A′| < μ for distinct A,A′ ∈ A. Our aim here is to study χCF(λ, κ, μ) = sup{χCF(A) : A is a (λ, κ, μ)-system} for λ ≥ κ ≥ μ, actually restricting ourselves to λ ≥ ω and μ ≤ ω. For instance, we prove that • for any limit cardinal κ (or κ = ω) and integers n ≥ 0, k > 0, GCH implies χCF(κ , t, k+1) =  κ+(n+1−i) if i · k < t ≤ (i+ 1) · k , i = 1, ..., n; κ if (n+ 1) · k < t ; • if λ ≥ κ ≥ ω > d > 1 , then λ < κ implies χCF(λ, κ, d) < ω and λ ≥ iω(κ) implies χCF(λ, κ, d) = ω ; • GCH implies χCF(λ, κ, ω) ≤ ω2 for λ ≥ κ ≥ ω2 and V=L implies χCF(λ, κ, ω) ≤ ω1 for λ ≥ κ ≥ ω1 ; • the existence of a supercompact cardinal implies the consistency of GCH plus χCF(אω+1, ω1, ω) = אω+1 and χCF(אω+1, ωn, ω) = ω2 for 2 ≤ n ≤ ω ; • CH implies χCF(ω1, ω, ω) = χCF(ω1, ω1, ω) = ω1, while MAω1 implies χCF(ω1, ω, ω) = χCF(ω1, ω1, ω) = ω . Date: March, 2010. 2000 Mathematics Subject Classification. 03E35, 03E05.