Transversal numbers of stacked spheres

A stacked d-sphere S is the boundary complex of a stacked (d+1)-ball, which is obtained by taking cone over a free d-face repeatedly from a (d+1)-simplex. A stacked sphere S is called linear if every cone is taken over a face added in the previous step. In this paper, we study the transversal ratio of facets of stacked d-spheres, which is the minimum proportion of vertices needed to cover all facets. Briggs, Dobbins and Lee showed that the transversal ratio of a stacked d-sphere is bounded above by 2d+2+o(1) and can be as large as 2d+3. We improve the lower bound by constructing linear stacked d-spheres with transversal ratio 63d+8 and general stacked d-spheres with transversal ratio 2d+3(d+2)2. Finally, we show that 63d+8 is optimal for linear stacked 2-spheres, that is, the transversal ratio is at most 37+o(1) for linear stacked 2-spheres.