2012/07/09 by Lajos Soukup, Soukup, Lajos
Mathematics · Computer Science · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1207.1971
Using Shelah's revised GCH theorem we prove that if mu<bethomega <= lambda\nare cardinals, then every mu-almost disjoint subfamily B of\n[lambda]bethomega is essentially disjoint, i.e. for each b from B there is\na subset f(b) of b of size < |b| such that the family b-f(b) b in B is\ndisjoint. We also show that if mu<=kappa<=lambda, and kappa is infinite, and\n(x) every mu-almost disjoint subfamily of [lambda]kappa is essentially\ndisjoint, then (xx) every mu-almost disjoint family B of subsets of lambda with\n|b|>=kappa for all b from B has a conflict-free colorings with kappa colors.\nPutting together these results we obtain that if mu<bethomega<=lambda, then\nevery mu-almost disjoint family B of subsets of lambda with |b|>=bethomega for\nall b from B has a conflict-free colorings with bethomega colors. To yield the\nabove mentioned results we also need to prove a certain compactness theorem\nconcerning singular cardinals.\n