论文标题
可定义的$（ω，2）$ - 定理VC代码密度小于$ 2 $的家庭定理
Definable $(ω, 2)$-theorem for families with VC-codensity less than $2$
论文作者
论文摘要
令$ \ Mathcal {s} $为VC代码密度小于$ 2 $的家族。我们证明，如果$ \ nathcal {s} $具有$（ω，2）$ - 属性（对于任何无限的无限套件，$ \ nathcal {s} $，至少$ 2 $相交的$ 2 $），则可以将$ \ nathcal {s} $分配为一定程度上的许多子families，每个属性分别是有限的。如果$ \ Mathcal {s} $在某些一阶结构中可以定义，则可以选择这些亚家族。这是对模型理论和Alon-Kleitman-Matoušek$（P，Q）$（P，Q）$的CASE $ Q = 2 $（P，Q）$ Q = 2 $的加强。
Let $\mathcal{S}$ be a family of sets with VC-codensity less than $2$. We prove that, if $\mathcal{S}$ has the $(ω, 2)$-property (for any infinitely many sets in $\mathcal{S}$, at least $2$ among them intersect), then $\mathcal{S}$ can be partitioned into finitely many subfamilies, each with the finite intersection property. If $\mathcal{S}$ is definable in some first-order structure, then these subfamilies can be chosen definable too. This is a strengthening of the case $q=2$ of the definable $(p,q)$- conjecture in model theory and of the Alon-Kleitman-Matoušek $(p,q)$-theorem in combinatorics.
