The Erdos-Rado arrow for singular cardinals

Todorcevic关于Rado猜想的难题

Todorcevic's Problem on Rado's Conjecture

本文证明了Rado猜想在所有正则基数上一致成立，从而回答了Todorčević在2024年Mostowski讲座中提出的关于两个连续基数上Rado猜想一致性的问题。

Rado猜想与随机代数

Rado's Conjecture and the random algebra

本文研究Rado猜想与随机代数的一致性，证明Rado猜想与$2^\omega=\omega_2$及Cichon图中随机代数对应的基数不变量一致，通过观察随机代数不特化非特殊树并构造新的Mitchell力迫得到结果。

Identities on cardinals less than aleph_omega

Let kappa be an uncountable cardinal and the edges of a complete graph with kappa vertices be colored with aleph_0 colors. For kappa >2^{aleph_0} the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if kappa <= 2^{aleph_0}, then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when kappa <= 2^{aleph_0}. In particular, we are concerned with the case aleph_1 <= kappa <= aleph_omega

数学中小基数的紧致性：原理、后果与局限性

Compactness for small cardinals in mathematics: principles, consequences, and limitations

本文讨论小正则基数（如ω_n、ℵ_{ω+1}等）的紧致性原理，分为逻辑原理（树性质）和数学原理（图、群、拓扑空间的紧致性），重点研究这些原理在力迫扩张中的不可破坏性，并观察传统问题（如Suslin假设、Whitehead猜想）与ω_2处最强紧致性形式的独立性。

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…