论文标题
VC饱和设置系统
VC-saturated set systems
论文作者
论文摘要
众所周知的sauer引理指出，$ \ mathcal {f} \ subseteq 2^{[n]} $ of vc-dimension of vc-dimension of $ d $最多有$ \ sum_ {i = 0}^d \ binom {i = 0}^d \ binom {n} {n} {i} $。我们获得随机和显式构造，以证明相应的饱和数，即具有VC-Dimension $ d \ ge 2 $的最小最大家族的大小，最多是$ 4^{d+1} $，因此独立于$ n $。
The well-known Sauer lemma states that a family $\mathcal{F}\subseteq 2^{[n]}$ of VC-dimension at most $d$ has size at most $\sum_{i=0}^d\binom{n}{i}$. We obtain both random and explicit constructions to prove that the corresponding saturation number, i.e., the size of the smallest maximal family with VC-dimension $d\ge 2$, is at most $4^{d+1}$, and thus is independent of $n$.
