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

Research paper by **András Hajnal, István Juhász, Lajos Soukup, Zoltán Szentmiklóssy**

Indexed on: **01 Apr '10**Published on: **01 Apr '10**Published in: **Mathematics - Logic**

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#### 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).