Infinite Asymptotic Combinatorics
نویسنده
چکیده
The following combinatorial theorems, some of which were known for every finite n in all infinite structures, are proved in ZFC for every infinite cardinal ν in all sufficiently large structures. (a) A new extension of Miller’s theorem [18]. (b) An upper bound of ρ on the list-conflict-free number of ρ-uniform families of sets which satisfy C(ρ, ν) for cardinals ν and ρ ≥ iω(n). (c) An upper bound of iω(ν) on the coloring number of a graph with list-chromatic number ν. (d) An extension to arbitrarily large cardinals of Komjáth’s comparison theorem [15] for א0-uniform families of sets. (e) The extensions of Miller’s theorem which were proved with the GCH by Erdős and Hajnal in the 1960s, and by Komjáth in the 1980s [2, 3, 14] and with the weaker axiom A(ν, ρ) by Hajnal, Juhász and Shelah [10]. The proofs rely on a consequence of Shelah’s theorem in cardinal arithmetic [23], by which every infinite cardinal satisfies with all sufficiently larger cardinals certain useful arithmetic relations that generalize the relation of finite cardinals to infinite ones.
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تاریخ انتشار 2012