你能去掉Komjath的不可达基数吗？
Can You Take Komjath's Inaccessible Away?
AI总结：
本文比较Kurepa树与Aronszajn树，通过序数步方法将相关相容性结论的大基数假设从两个不可达基数降至一个，还得到无需大基数的相容性结果及MA相关推论，并证明了Todorcevic ρ函数的新引理。
摘要：
本文旨在比较Kurepa树与Aronszajn树，并分析大基数假设对这一比较的影响。借助序数上的步方法（walks on ordinals），我们将证明：若存在一个不可达基数，则ZFC与“存在一棵Kurepa树且每棵Kurepa树都包含一棵Aronszajn子树”的命题相容。这一结果强于Komjath定理，后者从两个不可达基数出发证明了相同的相容性结论。此外，我们证明ZFC与以下命题相容：存在一棵Kurepa树$T$，使得若$U \subset T$是继承$T$的序的Kurepa树，则$U$具有一棵Aronszajn子树，该定理无需大基数假设。我们的最后一个定理可直接推出：假设$\textrm{MA}_{ω_2}$成立且$ω_2$在$\textsc{L}$中不是Mahlo基数，则存在一棵满足“每个Kurepa子集都有Aronszajn子树”性质的Kurepa树。我们的工作还证明了一个关于Todorcevic的$\rho$函数的新引理，该引理可能在其他场景中有用。
英文摘要：
In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains an Aronszajn subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem that asserts the same consistency from two inaccessible cardinals. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree $T$ such that if $U \subset T$ is a Kurepa tree with the inherited order from $T$, then $U$ has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: assume $\textrm{MA}_{ω_2}$ holds and $ω_2$ is not a Mahlo cardinal in $\textsc{L}$. Then there is a Kurepa tree with the property that every Kurepa subset has an Aronszajn subtree. Our work entails proving a new lemma about Todorcevic's $ρ$ function which might be useful in other contexts.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
