For an integer \(n\ge 2\), Ramsey Choice\(\mathsf {RC}_{n}\) 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 \(\mathsf {C}_{n}^{-}\) 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 \(n=2,3,4\), \(\mathsf {RC}_{n}\rightarrow \mathsf {C}_{n}^{-}\). However, the question of whether or not \(\mathsf {RC}_{n}\rightarrow \mathsf {C}_{n}^{-}\) for \(n\ge 5\) is still open. In general, for distinct \(m,n\ge 2\), not even the status of “\(\mathsf {RC}_{n}\rightarrow \mathsf {C}_{m}^{-}\)” or “\(\mathsf {RC}_{n}\rightarrow \mathsf {RC}_{m}\)” is known. In this paper, we provide partial answers to the above open problems and among other results, we establish the following:

1. 1. For every integer \(n\ge 2\), if \(\mathsf {RC}_{i}\) is true for all integers i with \(2\le i\le n\), then \(\mathsf {C}_{i}^{-}\) is true for all integers i with \(2\le i\le n\).
2. 2. If \(m,n\ge 2\) are any integers such that for some prime p we have \(p\not \mid m\) and \(p\mid n\), then in \(\mathsf {ZF}\): \(\mathsf {RC}_{m}\nrightarrow \mathsf {RC}_{n}\) and \(\mathsf {RC}_{m}\nrightarrow \mathsf {C}_{n}^{-}\).
3. 3. For \(n=2,3\), \(\mathsf {RC}_{5}\)\(+\)\(\mathsf {C}_{n}^{-}\) implies \(\mathsf {C}_{5}^{-}\), and \(\mathsf {RC}_{5}\) implies neither \(\mathsf {C}_{2}^{-}\) nor \(\mathsf {C}_{3}^{-}\) in \(\mathsf {ZF}\).
4. 4. For every integer \(k\ge 2\), \(\mathsf {RC}_{2k}\) 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 \(\mathsf {ZF}\) (for any \(k\in \omega \backslash \{0,1\}\)). In particular, \(\mathsf {RC}_{6}\) strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.
5. 5. The Chain-AntiChain Principle (“every infinite partially ordered set has either an infinite chain or an infinite anti-chain”) implies neither \(\mathsf {RC}_{n}\) nor \(\mathsf {C}_{n}^{-}\) in \(\mathsf {ZF}\), for every integer \(n\ge 2\).

中文翻译：

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关于n元集的无限族的Ramsey选择和局部选择

对于整数\（n \ ge 2 \），拉姆齐选择\（\ mathsf {RC} _ {n} \）是弱选择原则：“每个无限集x都有一个无限子集y，使得\（[y] ^ {n} \）（y的所有n个元素子集的集合）具有选择函数”，而\（\ mathsf {C} _ {n} ^ {-} \）是弱选择原则“每个无限大的族n个元素集合的一个具有选择函数“”的无限子族。1995年，黑山共和国证明对于\（n = 2,3,4 \），\（\ mathsf {RC} _ {n} \ rightarrow \ mathsf {C} _ {n} ^ {-} \）。但是，是否\（\ mathsf {RC} _ {N} \ RIGHTARROW \ mathsf {C} _ {N} ^ { - } \）为\（N \ GE 5 \）仍然是开放的。通常，对于不同的\（m，n \ ge 2 \），甚至没有“ \（\ mathsf {RC} _ {n} \ rightarrow \ mathsf {C} _ {m} ^ {-} \）的状态”或“ \（\ mathsf {RC} _ {n} \ rightarrow \ mathsf {RC} _ {m} \） ”已知。在本文中，我们为上述未解决的问题提供了部分答案，并在其他结果中建立了以下内容：

1. 1. 对于每个整数\（n \ ge 2 \），如果\（\ mathsf {RC} _ {i} \）对于所有带有\（2 \ le i \ le n \）的整数i为true  ，则\（ \ mathsf {C} _ {i} ^ {-} \）对于带有\（2 \ le i \ le n \）的所有整数i都是正确的 。
2. 2. 如果\（m，n \ ge 2 \）是任何整数，使得对于某些素数p我们具有\（p \ not \ mid m \）和\（p \ mid n \），则位于\（\ mathsf {ZF} \）：\（\ mathsf {RC} _ {m} \ nrightarrow \ mathsf {RC} _ {n} \）和\（\ mathsf {RC} _ {m} \ nrightarrow \ mathsf {C} _ {n} ^ {-} \）。
3. 3. 对于\（n = 2,3 \），\（\ mathsf {RC} _ {5} \）\（+ \）\（\ mathsf {C} _ {n} ^ {-} \）表示\ （\ mathsf {C} _ {5} ^ {-} \）和\（\ mathsf {RC} _ {5} \）都不暗示\（\ mathsf {C} _ {2} ^ {-} \）也不是\（\ mathsf {ZF} \）中的\（\ mathsf {C} _ {3} ^ {- } \）。
4. 4. 对于每个整数\（k \ ge 2 \），\（\ mathsf {RC} _ {2k} \）表示“每个无限长的线性k个元素集可有部分Kinna–Wagner选择函数”，并且在\（\ mathsf {ZF} \）中，后一种含义是不可逆的（对于任何\（k \ in \ omega \反斜杠\ {0,1 \} \））。特别地，\（\ mathsf {RC} _ {6} \）严格暗示“每个3元素集的无限线性可排序族都具有部分选择函数”。
5. 5. 该链式反链原理（“每无限部分有序集合具有任一的无限链或无限抗链”）意味着既不\（\ mathsf {RC} _ {N} \）也不\（\ mathsf {C} _ {n} ^ {-} \）在\（\ mathsf {ZF} \）中，对于每个整数\（n \ ge 2 \）。