In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: k$k$-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian. In this coloring, the set S(v)$S(v)$ of colors used by edges incident to a vertex v$v$ does not intersect S(u)$S(u)$ on more than k$k$ colors when u$u$ and v$v$ are adjacent. We provide some sharp upper and lower bounds for χ′k-int${\chi }_{k\text{-int}}^{\prime }$ for several classes of graphs. For l$l$-degenerate graphs we prove that χ′k-int(G)≤(l+1)Δ−l(k−1)−1${\chi }_{k\text{-int}}^{\prime }(G)\le (l+1)\mathrm{\Delta }-l(k-1)-1$. We improve this bound for subcubic graphs by showing that χ′2-int(G)≤6${\chi }_{2\text{-int}}^{\prime }(G)\le 6$. We show that calculating χ′k-int(Kn)${\chi }_{k\text{-int}}^{\prime }(K_n)$ for arbitrary values of k$k$ and n$n$ is related to some problems in combinatorial set theory and we provide bounds that are tight for infinitely many values of n$n$. Furthermore, for complete bipartite graphs we prove that χ′k-int(Kn,m)=⌈mnk⌉${\chi }_{k\text{-int}}^{\prime }(K_{n,m})=\lceil \frac{mn}{k}\rceil$. Finally, we show that computing χ′k-int(G)${\chi }_{k\text{-int}}^{\prime }(G)$ is NP-complete for every k≥1$k\ge 1$.