We prove that every finite set of homothetic copies of a given 1 compact and convex body in the plane can be colored with four colors 2 so that any point covered by at least two copies is covered by two copies 3 with distinct colors. This generalizes a previous result from Smorodinsky 4 (SIAM J. Disc. Math. 2007). Then we show that for any k ≥ 2, every 5 three-dimensional hypergraph can be colo...

The adaptable choosability of a multigraph G $G$ , denoted ch ( ) ${\text{ch}}_{a}(G)$ is the smallest integer k $k$ such that any edge labelling, τ $\tau $ and assignment lists size to vertices permits list colouring, σ $\sigma there no e = u v $e=uv$ where (e)=\sigma (u)=\sigma (v)$ . Here we show for with maximum degree Δ ${\rm{\Delta }}$ cycles length 3 or 4, ≤ 2 + o 1 ∕ ln ${\text{ch}}_{a}...