Transversal numbers of stacked spheres

Abstract

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 [Formula presented]. We improve the lower bound by constructing linear stacked d-spheres with transversal ratio [Formula presented] and general stacked d-spheres with transversal ratio [Formula presented]. Finally, we show that [Formula presented] is optimal for linear stacked 2-spheres, that is, the transversal ratio is at most [Formula presented]+o(1) for linear stacked 2-spheres. © 2024 Elsevier B.V.