Infinite asymptotic combinatorics

Shelah's work in cardinal arithmetic revealed that every infinite cardinal satisfies with every sufficiently larger cardinal arithmetic relations that finite cardinals satisfy with all infinite ones. Such relations are used here to prove in ZFC combinatorial theorems about infinite cardinals $\nu$ in sufficiently large structures, analogously to what is often proved for finite $n$ in all infinite structures. The following results are proved. (a) An extension of Miller's theorem \cite{miller}. (b) An upper bound of $\rho^+$ on the list-conflict-free number of $\rho$-uniform families of sets which satisfy $C(\rho^+,\nu)$ for cardinals $\nu$ and $\rho\ge\beth_\om(n)$. (c) An upper bound of $\beth_\om(\nu)$ on the coloring number of a graph with list-chromatic number $\nu$. (d) an extension to arbitrarily large cardinals Komj\'aths comparison theorem \cite{komcomp} for $\aleph_0$-uniform families sets. (e) Additional axioms are eliminated from extensions of Miller's theorem \cite{miller} which were proved in the 1960 by Erd\H os and Hajnal, and in the 1980s by Kom\'ath \cite{eh,eh1,komclose} with the GCH and which were re-proved in greater generality by Hajnal, Juh\'asz and Shelah \cite{hjs} from a additional axiom much weaker than the GCH.