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- W4376654340 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 number of facets of stacked $d$-spheres, denoted by $tau(S)$, which is the minimum number of vertices intersecting with all facets. Briggs, Dobbins and Lee showed that the transversal ratio of a stacked $d$-sphere is bounded above by $frac{2}{d+2}+o(1)$ and can be as large as $frac{2}{d+3}$. We improve the lower bound by constructing linear stacked $d$-spheres with transversal ratio $frac{6}{3d+8}$ and general stacked $d$-spheres with transversal ratio $frac{2d+3}{(d+2)^2}$. Finally, we show that $frac{6}{3d+8}$ is optimal for linear stacked $2$-spheres, that is, the transversal ratio is at most $frac{3}{7} + o(1)$ for linear stacked $2$-spheres." @default.
- W4376654340 created "2023-05-17" @default.
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- W4376654340 date "2023-05-15" @default.
- W4376654340 modified "2023-09-29" @default.
- W4376654340 title "Transversal numbers of stacked spheres" @default.
- W4376654340 doi "https://doi.org/10.48550/arxiv.2305.08716" @default.
- W4376654340 hasPublicationYear "2023" @default.
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