On Woodall's interval problem

For each k > 1, a sequence of 4(k ? 1) ? [log₂ (k ? 1)] sets (layers) of disjoint intervals in $\text{R}$ is constructed, such that every interval overlaps at least one interval in each preceding layer and no point of $\text{R}$ belongs to more than k layers. This improves a construction of Woodall. The approach is by way of the related “saturation” or “scope” problem.