Journal: Archive for Mathematical Logic

Abbreviation

Arch. math. log.

Publisher

Springer

Journal Volumes

ISSN

0933-5846
1432-0665

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2018 - 201912020 - 20232
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Publications 1 - 3 of 3
  • Some implications of Ramsey Choice for families of n-element sets
    Item type: Journal Article
    Halbeisen, Lorenz; Schumacher, Salome (2023)
    Archive for Mathematical Logic
    For n∈ω, the weak choice principle RCn is defined as follows: For every infinite set X there is an infinite subset Y⊆X with a choice function on [Y]n:={z⊆Y:|z|=n}. The choice principle C−n states the following: For every infinite family of n-element sets, there is an infinite subfamily G⊆F with a choice function. The choice principles LOC−n and WOC−n are the same as C−n, but we assume that the family F is linearly orderable (for LOC−n) or well-orderable (for WOC−n). In the first part of this paper, for m,n∈ω we will give a full characterization of when the implication RCm⇒WOC−n holds in ZF. We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that RC5⇒LOC−5 and that RC6⇒C−3, answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that RC6⇒C−9 and that RC7⇒LOC−7.
  • On Ramsey choice and partial choice for infinite families of n-element sets
    Item type: Journal Article
    Halbeisen, Lorenz; Tachtsis, Eleftherios (2020)
    Archive for Mathematical Logic
    For an integer n≥ 2 , Ramsey ChoiceRCn is the weak choice principle “every infinite setxhas an infinite subset y such that[y] n (the set of alln-element subsets of y) has a choice function”, and Cn- is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for n= 2 , 3 , 4 , RCn→Cn-. However, the question of whether or not RCn→Cn- for n≥ 5 is still open. In general, for distinct m, n≥ 2 , not even the status of “RCn→Cm-” or “RCn→ RCm” is known. In this paper, we provide partial answers to the above open problems and among other results, we establish the following: 1.For every integer n≥ 2 , if RCi is true for all integers i with 2 ≤ i≤ n, then Ci- is true for all integers i with 2 ≤ i≤ n. 2.If m, n≥ 2 are any integers such that for some prime p we have p∤ m and p∣ n, then in ZF: RCm↛ RCn and RCm↛Cn-. 3.For n= 2 , 3 , RC5+ Cn- implies C5-, and RC5 implies neither C2- nor C3- in ZF. 4.For every integer k≥ 2 , RC2 k implies “every infinite linearly orderable family of k-element sets has a partial Kinna–Wagner selection function” and the latter implication is not reversible in ZF (for any k∈ ω\ { 0 , 1 }). In particular, RC6 strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”. 5.The Chain-AntiChain Principle (“every infinite partially ordered set has either an infinite chain or an infinite anti-chain”) implies neither RCn nor Cn- in ZF, for every integer n≥ 2. © 2019 Springer.
  • A weird relation between two cardinals
    Item type: Journal Article
    Halbeisen, Lorenz (2018)
    Archive for Mathematical Logic
Publications 1 - 3 of 3