Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions

Abstract

Let $f_{r}(d,s_{1},\ldots,s_{r})$ be the least $N$ such that every $N$-point set $P\subseteq \mathbb{R}^{d}$ has an $r$-partition $P=P_{1}\sqcup\cdots\sqcup P_{r}$ with the following property: whenever $C_{i}\supseteq P_{i}$ is a union of at most $s_{i}$ convex sets, one has $\bigcap_{i=1}^{r}C_{i}\ne\emptyset$. A recent breakthrough of Alon and Smorodinsky established the first effective upper bounds $f_{r}(d,s,\ldots,s)\le Cdr^{2}s^{r}\log r\log(es^{r})$ for this problem. We obtain an asymptotically sharp lower bound by proving $f_r(d,s,\ldots,s)\ge c(d-r+2)s^r\log(s+1)$ for every $d\ge r+2$, which shows that $f_r(d,s,\ldots,s)=Θ_{d,r}(s^r\log s)$ for every fixed $d\ge r+2$. We also prove the general lower bound $f_r(d,s,\ldots,s)>s^{\min\{d,r\}}$. On the other hand, we develop a local counting argument to show that $f_r(d,s,\ldots,s)\le C_{d}rs^r\log(ers^r)$ and $f_r(d,s,\ldots,s)\le C_{d}r^{d+2}s^{d+1}\log(ers)$ whenever $r\ge d+1$, improving the upper bound of Alon and Smorodinsky. We also study two colored analogues. The direct Bárány--Larman-type extension, in which one seeks $r$ disjoint rainbow sets chosen from $d+1$ color classes, fails as soon as two convex pieces are allowed. Nevertheless, we identify the correct colored formulation and prove a complete transversal theorem with quantitative bounds, which was also independently obtained by Keller and Smorodinsky.