Let $N$ be a finite set and $D=\{K_1,\ldots, K_m\}$ be a set containing subsets of $N$. It is given that every element of $N$ is covered by at least $\gamma$ elements of $D$. I am trying to find estimation for minimal covering of $N$ using subsets in $D$.
I have seen this result: There exists a covering with no more than $\frac{(1+ln(\gamma)|N|)}{\gamma}+1$ elements but can't find the proof. I am not sure if it is correct.
Can anyone help on this problem?
If $N= \{1,2,3,...,n\}$ and $D= \{\{1\},\{2\},...,\{n\}\}$ then $\gamma = 1$. Then $\frac{(1+ln(\gamma)|N|)}{\gamma}+1=2$. But there is no 2-cover of $N$? Or did I misunderstand the problem?