Ideals, Cohen sets and consistent extensions of the Erdős-Dushnik-Miller Theorem

Erdős-Dushnik-Miller 定理的三种形式

Three forms of the Erdős-Dushnik-Miller Theorem

本文在无选择公理的集合论中研究 Erdős-Dushnik-Miller 定理，证明其存在三个不等价形式，并刻画它们在弱选择原则演绎层级中的位置。

Erdös–Dushnik–Miller定理的无选择证明

A choice-free proof of the Erdős--Dushnik--Miller theorem

本文在不含选择公理的ZF集合论中，给出Erdös–Dushnik–Miller定理的纯组合证明，避免了元数学考量。

关于无选择公理的Erdos--Dushnik--Miller定理

On Erdos--Dushnik--Miller theorem without AC

本文在无选择公理的集合论中，研究了限制于不可数顶点集的Erdos-Dushnik-Miller定理在弱选择形式层级中的位置，并部分回答了Lajos Soukup提出的问题。

Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,ρ)$ in which $n$ is finite and $ρ$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(ν,ρ)$ in which $ν$ is arbitrary and $ρ\ge \beth_\om(ν)$. The proof uses a new general method that is based on Shelah's revises Generalized Continuum Hypothesis theorem. Upper bounds on conflict-free coloring numbers of families of sets and a general comparison theorem follow as corollaries of the main theorem. Other corollaries eliminate the use of additional axioms from splitting theorems due to E…

An infinite combinatorial statement with a poset parameter

We introduce an extension, indexed by a partially ordered set P and cardinal numbers k,l, denoted by (k,l)-->P, of the classical relation (k,n,l)--> r in infinite combinatorics. By definition, (k,n,l)--> r holds, if every map from the n-element subsets of k to the subsets of k with less than l elements has a r-element free set. For example, Kuratowski's Free Set Theorem states that (k,n,l)-->n+1 holds iff k is larger than or equal to the n-th cardinal successor l^{+n} of the infinite cardinal k. By using the (k,l)-->P framework, we present a self-contained proof of the first author's result th…