کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4665612 | 1633819 | 2015 | 19 صفحه PDF | دانلود رایگان |
Miller's 1937 splitting theorem was proved for every finite n>0n>0 for all ρ-uniform families of sets in which ρ is infinite. A simple method for proving Miller-type splitting theorems is presented here and an extension of Miller's theorem is proved in ZFC for every cardinal ν for all ρ -uniform families in which ρ≥ℶω(ν)ρ≥ℶω(ν). The main ingredient in the method is an asymptotic infinitary Löwenheim–Skolem theorem for anti-monotone set functions.As corollaries, the use of additional axioms is eliminated from splitting theorems due to Erdős and Hajnal [1], Komjáth [7], Hajnal, Juhász and Shelah [4]; upper bounds are set on conflict-free coloring numbers of families of sets; and a general comparison theorem for ρ -uniform families of sets is proved, which generalizes Komjáth's comparison theorem for ℵ0ℵ0-uniform families [8].
Journal: Advances in Mathematics - Volume 269, 10 January 2015, Pages 707–725