Using Shelah's revised GCH theorem we prove that if mu<beth_omega <= lambda
are cardinals, then every mu-almost disjoint subfamily B of
[lambda]^{beth_omega} is essentially disjoint, i.e. for each b from B there is
a subset f(b) of b of size < |b| such that the family {b-f(b) b in B} is
disjoint.
We also show that if mu<=kappa<=lambda, and kappa is infinite, and (x) every
mu-almost disjoint subfamily of [lambda]^kappa is essentially disjoint, then
(xx) every mu-almost disjoint family B of subsets of lambda with |b|>=kappa for
all b from B has a conflict-free colorings with kappa colors.
Putting together these results we obtain that if mu<beth_omega<=lambda, then
every mu-almost disjoint family B of subsets of lambda with |b|>=beth_omega for
all b from B has a conflict-free colorings with beth_omega colors.
To yield the above mentioned results we also need to prove a certain
compactness theorem concerning singular cardinals.Comment: 10 pages, minor correction
