Choosing between incompatible ideals
Choosing between incompatible ideals
复制标题
在不相容的理想之间进行选择
DOI：
10.1016/j.ejc.2021.103349
复制
发表时间：
2021
影响因子：
1
通讯作者：
Larson, Paul B.
中科院分区：
文献类型：
--
作者：
Brian, Will;Larson, Paul B.
假设I和J是集合x上的真理想，如果I∪J不能生成真理想，我们说I和J是不相容的。等价于I和J是不相容的,如果有一些⊆X,这样∈我和∖∈J .如果B⊆X是在I∖J或J∖我,然后我们说B之间选择我和J .我们考虑以下Ramsey-theoretic问题:鉴于几双(I 1 J 1), (I, J 2),…,(我k, J k)不兼容的理念在一组X,找到一些⊆X之间选择尽可能多的这些成对的理想。主要定理是，对于每个n∈n，存在某个I (n)∈n，使得给定任意集合X上至少I (n)对不相容理想，存在某个A (X)在其中至少n个理想中进行选择。这个定理的证明主要分为两个步骤。第一步是识别极值组合中的一个（纯有限）问题，并证明我们的理想问题等价于这个组合问题。第二步是分析组合问题，以证明上面描述的数字I (n)存在，并为其设定界限。我们证明了1 2nlog2n−O (n)< I (n)< nlnn + O (n)。上界是通过考虑涉及超图的一个不同但密切相关的组合问题来证明的，这可能是独立的兴趣。我们还研究了该定理在一个条件收敛级数问题上的一些应用。
Suppose I and J are proper ideals on some set X. We say that I and J are incompatible if I∪ J does not generate a proper ideal. Equivalently, I and J are incompatible if there is some A⊆ X such that A∈ I and X∖ A∈ J. If some B⊆ X is either in I∖ J or in J∖ I, then we say that B chooses between I and J. We consider the following Ramsey-theoretic problem: Given several pairs (I 1, J 1),(I 2, J 2),…,(I k, J k) of incompatible ideals on a set X, find some A⊆ X that chooses between as many of these pairs of ideals as possible. The main theorem is that for every n∈ N, there is some I (n)∈ N such that given at least I (n) pairs of incompatible ideals on any set X, there is some A⊆ X choosing between at least n of them. This theorem is proved in two main steps. The first step is to identify a (purely finitary) problem in extremal combinatorics, and to show that our problem concerning ideals is equivalent to this combinatorial problem. The second step is to analyze the combinatorial problem in order to show that the number I (n) described above exists, and to put bounds on it. We show 1 2 n log 2 n− O (n)< I (n)< n ln n+ O (n). The upper bound is proved by considering a different but closely related combinatorial problem involving hypergraphs, which may be of independent interest. We also investigate some applications of this theorem to a problem concerning conditionally convergent series.
DOI：
--
发表时间：
2019
期刊：
The Crimean Karaim Bible
影响因子：
--
作者：
John Horton Conway Richard K Guy
通讯作者：
John Horton Conway Richard K Guy
三个条件收敛级数，但不是四个
影响因子：
0.9
作者：
W. Brian
通讯作者：
W. Brian
子系列号
影响因子：
0.6
作者：
Brendle Joerg;Brian Will;Hamkins Joel David
通讯作者：
Hamkins Joel David