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.

What is the maximum number of vertices that a centrally symmetric $2$-neighborly polytope of dimension $d$ can have? It is known that the answer does not exceed $2^d$. Here we provide an explicit construction showing that it is at least $2^{d-1}+2$.

A notion of an $i$-banner simplicial complex is introduced. For various values of $i$, these complexes interpolate between the class of flag complexes and the class of all simplicial complexes. Examples of simplicial spheres of an arbitrary dimension that are $(i+1)$-banner but not $i$-banner are constructed. It is shown that several theorems for flag complexes have appropriate $i$-banner analogues. Among them are (1) the codimension-$(i+j-1)$ skeleton of an $i$-banner homology sphere $\Delta$ is $2(i+j)$-Cohen--Macaulay for all $0\leq j\leq \dim\Delta+1-i$, and (2) for every $i$-banner simplicial complex $\Delta$ there exists a balanced complex $\Gamma$ with the same number of vertices as $\Delta$ whose face numbers of dimension $i-1$ and higher coincide with those of $\Delta$.