Transversal numbers of simplicial polytopes, spheres, and pure complexes

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 $d$-dimensional polytopes that could be considered as ``siblings'' of cyclic polytopes and show that the transversal ratios of such odd-dimensional polytopes are $2/5-o(1)$. The previous record for the transversal ratios of $(2k+1)$-polytopes was $1/(k+1)$. Finally, we construct infinite families of $3$-, $4$-, and $5$-dimensional simplicial spheres with transversal ratios converging to $4/7$, $1/2$, and $6/11$, respectively. The previous record was $11/21$, $2/5$, and $1/2$, respectively.