  Let $\mathcal{F}$ be a family of subsets of $[n]=\{1,\ldots,n\}$ and let $L$
be a set of nonnegative integers. The family $\mathcal{F}$ is
\emph{$L$-intersecting} if $|F\cap F'|\in L$ for every two distinct members
$F,F'\in\mathcal{F}$; and $\mathcal{F}$ is $k$-uniform if all its members have
the same size $k$. A large variety of problems and results in extremal set
theory concern on $k$-uniform $L$-intersecting families. Many attentions are
paid to finding the maximum size of a family among all $k$-uniform
$L$-intersecting families with prescribed $n,k$ and $L$. In this paper, from
another point of view, we propose and investigate the problem of estimating the
maximum size of a member in a family among all uniform $L$-intersecting
families with size $m$, here $n,m$ and $L$ are prescribed. Our results aim to
find out more precise relations of $n,m,k$ and $L$.
