Eprints of Rényi Institute

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

Hajnal, Andras and Juhász, Istvan and Soukup, Lajos and Szentmiklossy, Zoltan (2010) Conflict free colorings of (strongly) almost disjoint set-systems. (Submitted)

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Abstract

f:\cup\mathcal{A}\to {\rho} is called a conflict free coloring of the set-system \mathcal{A} (with {\rho} colors) if \forall A\in \mathcal{A}\,\, \exists\, {\zeta <{\rho}\, (\,|A\cap f^{-1}\{{\zeta}\}|=1\,). The conflict free chromatic number {\chi_{CF (\mathcal{A}) of \mathcal{A} is the smallest \rho for which \mathcal{A} admits a conflict free coloring with {\rho} colors. \mathcal{A} is a (\lambda,\kappa,\mu)-system if |\mathcal{A}| = \lambda, |A| = \kappa for all A \in \mathcal{A}, and \mathcal{A} is {\mu}-almost disjoint, i.e. |A\cap A'|<{\mu} for distinct A, A'\in \mathcal{A}.
Our aim here is to study {\chi_{CF}}(\lambda,\kappa,\mu) = \sup \{{\chi_{CF} (\mathcal{A}): \mathcal{A} \mbox{  is a } (\lambda,\kappa,\mu)\mbox{-system} \} for \lambda \ge \kappa \ge \mu, actually restricting ourselves to \lambda \ge \omega and \mu \le \omega. For instance, we prove that
\bullet for any limit cardinal  \kappa (or \kappa = \omega) and integers n \ge 0,\,k > 0, GCH implies {\chi_{CF}}(\kappa^{+n},t,k+1) = \kappa^{+(n+1-i)} if i\cdot k < t \le (i+1)\cdot k\,,i = 1,...,n, {\chi_{CF}}(\kappa^{+n},t,k+1)=\kappa if (n+1)\cdot k < t\,;
\bullet if \lambda \ge \kappa \ge \omega > d > 1\,, then \,\lambda < \kappa^{+\omega} implies {\chi_{CF} (\lambda,\kappa,d) < \omega and \lambda \ge \beth_\omega(\kappa)\, implies {\chi_{CF}}(\lambda,\kappa,d) = \omega\,;
\bullet GCH implies \,{\chi_{CF}}(\lambda,\kappa,\omega) \le \omega_2 for \lambda \ge \kappa \ge \omega_2,
\bullet V=L implies \,{\chi_{CF}}(\lambda,\kappa,\omega) \le  \omega_1 for \lambda \ge \kappa \ge \omega_1\,;
\bullet the existence of a supercompact cardinal implies the consistency of GCH plus {\chi_{CF}}(\aleph_{\omega+1},\omega_1,\omega)= \aleph_{\omega+1} and {\chi_{CF}}(\aleph_{\omega+1},\omega_n,\omega) = \omega_2 for 2 \le n \le \omega\, ;
\bullet CH implies \,{\chi_{CF}}(\omega_1,\omega,\omega) = {\chi_{CF}}(\omega_1,\omega_1,\omega) =\omega_1, while MA_{\omega_1} implies \,{\chi_{CF}}(\omega_1,\omega,\omega) = {\chi_{CF}}(\omega_1,\omega_1,\omega) =\omega\,.

Item Type:Research Paper
Keywords and phrases:coloring, conflict-free coloring, almost disjoint, essentially disjoint
Subjects:03E35 Consistency and independence results
Divisions:Research Divisions > Set theory and general topology
Last Modified:02 Apr 2010 11:16

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