Heterochromatic Higher Order Transversals for Convex Sets

For $0\leq k\leq d-1$, a $k$-flat in $\mathbb{R}^d$ is a $k$-dimensional affine subspace in $\mathbb{R}^d$. A set $T$ of $k$-flats in $\mathbb{R}^d$ is a {\em $k$-transversal} of a family $\mathcal{F}$ of subsets of $\mathbb{R}^d$ if every member of $\mathcal{F}$ intersects some $k$-flat in $T$. A family $\mathcal{F}$ of sets is said to satisfy $(p,q)$-property with respect to a set $T$ of $k$-flats if among every $p$ members of $\mathcal{F}$ at least $q$ of them can be hit by a single $k$-flat in $T$. Keller and Perles (SoCG 2022) proved an $(\aleph_0,k+2)$-theorem for $k$-transversals for families of $(r,R)$-fat convex sets in $\mathbb{R}^d$. They proved that any family $\mathcal{F}$ of $(r,R)$-fat sets in $\mathbb{R}^d$ satisfying $(\aleph_{0},k+2)$-property with respect to $k$-transversals can be hit by a finite number of $k$-flats. In this paper, we have extended the $(\aleph_0,k+2)$-theorem for $k$-transversals without the fatness assumption by introducing a notion, called \emph{$k$-growing sequence}. Moreover, we have proved a heterochromatic version of $(\aleph_0,k+2)$-theorem for $k$-transversals for families of both $(r,R)$-fat sets and more general convex sets. We have also proved a colorful generalization of the \emph{Helly-type} theorem for $k$-transversals of convex sets due to Aronov, Goodman and Pollack.