Abstract. We prove new upper and lower bounds on transversal numbers of several classes of simplicial complexes. Specifically, we establish an upper bound on the transversal numbers of pure simplicial complexes in terms of the number of vertices and the number of facets and then provide constructions of pure simplicial complexes whose transversal numbers come close to this bound. We introduce a new family of [Formula: see text]-dimensional polytopes that could be considered as “siblings” of cyclic polytopes and show that the transversal ratios of such odd-dimensional polytopes are [Formula: see text]. The previous record for the transversal ratios of [Formula: see text]-polytopes was [Formula: see text]. Finally, we construct infinite families of 3-, 4-, and 5-dimensional simplicial spheres with transversal ratios converging to [Formula: see text], [Formula: see text], and [Formula: see text], respectively. The previous record was [Formula: see text], [Formula: see text], and [Formula: see text], respectively.

We determine the minimum number of vertices needed to provide balanced triangulations of $\mathbb{S}^{d-2}$-bundles over $\mathbb{S}^1$. If $d$ is odd and the bundle is orientable, or $d$ is even and the bundle is nonorientable, the minimum number of vertices is $3d$; otherwise, it is $3d+2$. Similar results apply to all balanced simplicial complexes that triangulate homology manifolds with $\beta_1\neq0$ and $\beta_2=0$, where $\beta_i$'s are the Betti numbers, computed with coefficients in $\mathbb{Q}$.