$f\: \cup {\mathcal {A}}\to {\rho}$ is called a conflict free coloring of the set-system (with ρ colors) if $$\forall A\in {\mathcal {A}}\ \exists\, {\zeta}<{\rho} (|A\cap f^{-1}\{{\zeta}\}|=1).$$ The conflict free chromatic number $\operatorname {\chi _{\rm CF}}\, ({\mathcal {A}})$ of is the smallest ρ for which admits a conflict free coloring with ρ colors.
is a (λ,κ,μ)-system if , |A|=κ for all , and is μ-almost disjoint, i.e. |A∩A′|<μ for distinct . Our aim here is to study $$\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\mu) = \sup \{\operatorname {\chi _{\rm CF}}\, ({\mathcal {A}})\: {\mathcal {A}}\mbox{ is a } (\lambda,\kappa,\mu)\mbox{-system}\}$$ 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 $$\operatorname {\chi _{\rm CF}}\, (\kappa^{+n},t,k+1) =\begin{cases}\kappa^{+(n+1-i)}& \text{if \ } i\cdot k < t \le (i+1)\cdot k,\ i =1,\dots,n;\\[2pt]\kappa& \text{if \ } (n+1)\cdot k < t;\end{cases}$$
• if λ≧κ≧ω>d>1, then λ<κ +ω implies $\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,d) <\omega$ and λ≧ℶ ω (κ) implies $\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,d) = \omega$ ;
• GCH implies $\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\omega) \le \omega_{2}$ for λ≧κ≧ω 2 and V=L implies $\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\omega) \le \omega_{1}$ for λ≧κ≧ω 1;
• the existence of a supercompact cardinal implies the consistency of GCH plus $\operatorname {\chi _{\rm CF}}\,(\aleph_{\omega+1},\omega_{1},\omega)= \aleph_{\omega+1}$ and $\operatorname {\chi _{\rm CF}}\, (\aleph_{\omega+1},\omega_{n},\omega) = \omega_{2}$ for 2≦n≦ω;
• CH implies $\operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega,\omega) = \operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega_{1},\omega) = \omega_{1}$ , while implies $\operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega,\omega) = \operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega_{1},\omega) = \omega$ .