The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree $\delta \ge {\ln }^{25}\mathrm{\Delta }$, the following holds: for every assignment L of lists of colours to the edges of G, such that $|L(e)|\ge (1+o(1))\cdot \max \{\mathrm{deg}(u),\mathrm{deg}(v)\}$ for each edge $e=uv$, there is an L-edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.

20 世纪 70 年代著名的列表着色猜想指出，对于每个图G ， G的色指数等于其列表色指数。1996 年，卡恩在一篇开创性的论文中证明了列表着色猜想渐近成立。我们的主要结果是卡恩定理的局部推广。更准确地说，我们表明，对于具有足够大的最大度 Δ 和最小度的图G$\delta \ge {\mathrm{因}}^{25}\mathrm{\Delta }$，以下内容成立：对于每个将颜色列表L分配给G的边缘，使得$|L（e）|\ge （1+哦（1））\cdot \mathrm{最大限度}\{\mathrm{度}（你）,\mathrm{度}（v）\}$对于每条边$e=你v$，有G的L边着色。此外，卡恩证明列表着色猜想对于线性、k均匀超图渐近成立，最近莫洛伊将卡恩的原始结果推广到对应着色及其超图泛化。我们通过显示同时暗示我们所有结果的加权版本来证明所有这些概括的本地版本。