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

Abstract

A set-system $X$ is a $(\lambda, \kappa,\mu)$-system iff $|X|=\lambda$, $|x|=\kappa$ for each $x\in X$, and $X$ is $\mu$-almost disjoint. We write $[\lambda, \kappa, \mu] -> \rho$ iff every $(\lambda, \kappa,\mu)$-system has a "conflict free coloring with $\rho$ colors", i.e. there is a coloring of the elements of $\cup X$ with$\rho$ colors such that for each element $x$ of $X$ there is a color $\xi<\rho$ such that exactly one element of $x$ has color $\xi$. Our main object of study is the relation $[\lambda, \kappa, \mu] -> \rho$. We give full description of this relation when $\kappa$ is finite. We also show that if $d$ is a natural number then $[\lambda,\kappa,d]-> \omega$ always holds. Under GCH we prove that $[\lambda,\kappa,\omega]-> \omega_2$ holds for $\kappa>\omega_1$, but the relation $[\lambda,\kappa,\omega]-> \omega_1$ is independent (modulo some large cardinals).