  A $q$-graph $H$ on $n$ vertices is a set of vectors of length $n$ with all
entries from $\{0,1,\dots,q\}$ and every vector (that we call a $q$-edge)
having exactly two non-zero entries. The support of a $q$-edge $\mathbf{x}$ is
the pair $S_{\mathbf{x}}$ of indices of non-zero entries. We say that $H$ is an
$s$-copy of an ordinary graph $F$ if $|H|=|E(F)|$, $F$ is isomorphic to the
graph with edge set $\{S_{\mathbf{x}}:\mathbf{x}\in H\}$, and whenever $v\in
e,e'\in E(F)$, the entries with index corresponding to $v$ in the $q$-edges
corresponding to $e$ and $e'$ sum up to at least $s$. E.g., the $q$-edges
$(1,3,0,0,0), (0,1,0,0,3)$, and $(3,0,0,0,1)$ form a 4-triangle.
  The Tur\'an number $\mathrm{ex}(n,F,q,s)$ is the maximum number of $q$-edges
that a $q$-graph $H$ on $n$ vertices can have if it does not contain any
$s$-copies of $F$. In the present paper, we determine the asymptotics of
$\mathrm{ex}(n,F,q,q+1)$ for many graphs $F$.
