Transversal numbers of simplicial polytopes, spheres, and pure complexes
Isabella Novik, Hailun Zheng
TL;DR
The paper advances the understanding of transversal numbers for higher-dimensional combinatorial objects by deriving a general upper bound $T(\Delta) \le n+1-\frac{1}{e}\,nm^{-1/d}$ for pure complexes, and by constructing families that nearly attain this bound. It establishes new lower bounds on transversal ratios, notably $\tau^P_{2k+1} \ge 2/5$ for odd-dimensional polytopes and $\tau^S_4 \ge 4/7$, $\tau^S_5 \ge 1/2$, $\tau^S_6 \ge 6/11$ for spheres, using Gale’s evenness condition, Shemer sewing, and bistellar retriangulations. The authors introduce siblings of cyclic polytopes via sewing to derive $\tau^P_{2k+1} \ge 2/5$ for all $k$, and they construct PL spheres in dimensions 3, 4, and 5 with progressively larger transversal ratios by strategic retriangulations, significantly improving prior records. Collectively, these techniques shed light on Turán-type problems for pure complexes and offer new tools and objects of independent interest in polytope and PL-topology contexts.
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 $d$-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.