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I have a range space $(X,R)$, were $R$ is a collection of subsets of $R$ and I have an upper bound $d$ to the VC-dimension of $(X,R)$. Suppose for simplicity that $X$ is finite. Given $\delta\in(0,1)$ and an integer $m>1$, I know that if I have a random sample $S$ from X of size $|S|=m$, then for $\varepsilon=\sqrt{\frac{d+\log(1/\delta)}{m}}$, we have $\Pr\left(\exists r\in R ~:~ \left|\frac{|r|}{|X|}-\frac{|r\cap S|}{|S|}\right|>\varepsilon \right)<\delta$.

Notice how $\varepsilon$ does not depend on the sizes of the members of $R$. I was wondering whether I can get a smaller $\varepsilon$ if I know that all members of $R$ have a specific size, or better, if I know that there is a $\phi\in(0,1)$ sucht that $\frac{|r|}{|X|}=\phi$ for all $r\in R$.

Thanks.

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No, you cannot. See Thm 2.1 of http://link.springer.com/article/10.1007%2FBF02187833 (it says that a random family gives an example, with appropriately chosen parameters).

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