论文标题
在$κ$ - 均匀的，但不超过$κ$ - $ - $ transitive排列组
On $κ$-homogeneous, but not $κ$-transitive permutation groups
论文作者
论文摘要
套装$ a $ a $的排列组$ g $是$κ$ -HOMESOUSIFF，用于所有$ x，y \ in [a]^κ$，带有$ | a \ setminus x | = | a \ setminus y | = | a | $ g $是任何注射函数$ f $的$ g $ in [a]^{a]^{\ le-beκ} $和$ | a \ setMinus dom（f）| = | = | a setminus ran（f） 给出部分答案。 （i）$λ<ω_Ω$，或 （ii）$ 2^ω<λ$，以及$μ^ω=μ^+$和$ \box_μ$，每$μ\leλ$ with $ω= cf（μ）<{μ} $或 （iii）我们的模型是通过在某些地面模型中添加$ω_1$ $ω_1$来获得的。 对于$κ>ω$，我们提供了一种构建大型$κ$均匀的（但不是$κ$传递置换组）的方法。使用此方法，我们表明存在$κ^+$ - 均质，但不存在$κ^+$ - $κ^{+n} $上的$κ^+$ - 对于每个无限的Cardinal $κ$和自然数量$ n \ ge 1 $提供的$ v = $ v = l $。
A permutation group $G$ on a set $A$ is $κ$-homogeneous iff for all $X,Y\in [A]^κ$ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. $G$ is $κ$-transitive iff for any injective function $f$ with $dom(f)\cup ran(f)\in [A]^{\le κ}$ and $|A\setminus dom(f)|=|A\setminus ran(f)|=|A|$ there is a $g\in G$ with $f\subset g$. Giving a partial answer to a question of P. M. Neumann we show that there is an $ω$-homogeneous but not $ω$-transitive permutation group on a cardinal $λ$ provided (i) $λ<ω_ω$, or (ii) $2^ω<λ$, and $μ^ω=μ^+$ and $\Box_μ$ hold for each $μ\leλ$ with $ω=cf(μ)<{μ}$, or (iii) our model was obtained by adding $ω_1$ many Cohen generic reals to some ground model. For $κ>ω$ we give a method to construct large $κ$-homogeneous, but not $κ$-transitive permutation groups. Using this method we show that there exists $κ^+$-homogeneous, but not $κ^+$-transitive permutation groups on $κ^{+n}$ for each infinite cardinal $κ$ and natural number $n\ge 1$ provided $V=L$.
