  In this paper we raise a variant of a classic problem in extremal graph
theory, which is motivated by a design of fractional repetition codes, a model
in distributed storage systems. For any feasible positive integers $d\geq 3$,
$n \geq 3$, and $k$, where $n-1 \leq k \leq \binom{n}{2}$, what is the minimum
possible number of vertices in a $d$-regular undirected graph whose subgraphs
with $n$ vertices contain at most $k$ edges? The goal of this paper is to give
the exact number of vertices for each instance of the problem and also to
provide some bounds for general values of $n$, $d$, and $k$. A few general
bounds with some exact values, for this Tur\'an-type problem, are given. We
present an almost complete solution for $3 \leq n \leq 5$.
