Conflict free colorings of (strongly) almost disjoint set-systems

A set-system $X$ is a $(λ, κ,μ)$-system iff $|X|=λ$, $|x|=κ$ for each $x\in X$, and $X$ is $μ$-almost disjoint. We write $[λ, κ, μ] -> ρ$ iff every $(λ, κ,μ)$-system has a "conflict free coloring with $ρ$ colors", i.e. there is a coloring of the elements of $\cup X$ with$ρ$ colors such that for each element $x$ of $X$ there is a color $ξ<ρ$ such that exactly one element of $x$ has color $ξ$. Our main object of study is the relation $[λ, κ, μ] -> ρ$. We give full description of this relation when $κ$ is finite. We also show that if $d$ is a natural number then $[λ,κ,d]-> ω$ always holds. Under GCH we prove that $[λ,κ,ω]-> ω_2$ holds for $κ>ω_1$, but the relation $[λ,κ,ω]-> ω_1$ is independent (modulo some large cardinals).

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