Edge-colouring graphs with local list sizes

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 $\delta \ge {\ln }^{25}\Delta$, the following holds: for every assignment L of lists of colours to the edges of G, such that $\left|L\right(e\left)\right|\ge (1+o(1\left)\right)\cdot \max \{\mathrm{deg}(u),\mathrm{deg}(v\left)\right\}$ 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.