Properties of closure operators in the plane
Abstract
We prove a general version of Radon's theorem for finite families $\mathcal F$ of sets in zero, one and twodimensional (pseudo)manifolds and graphs. The only requirement is that for each $\mathcal G\subseteq\mathcal F$ the intersection $\bigcap\mathcal G$ has bounded number of pathconnected components. As a consequence we obtain Helly's and Tverberg's theorems, fractional and colorful Helly theorems, existence of weak $\varepsilon$nets and $(p,q)$theorems. More precisely, if $\mathcal F$ is an intersectionclosed family in a topological space $X$ and $S\subseteq X$, we define its convex hull $\text{conv}_{\mathcal F}(S)$ as the smallest set in $\mathcal F$ that contains $S$. The Radon number $r(\mathcal F)$ is then the smallest integer $r$ such that every set $S\subseteq X$ of size $r$ can be split into two disjoint subsets with intersecting convex hulls. For every graph $G$ with $n$ vertices we construct a polynomial $p_G(x)$ satisfying the following three conditions. 1) If $n=1$, $p_G(x)=1$. 2) If $n\geq 2$, the degree of $p_G(x)$ is at most $2n3$. 3) If the graph $G$ does not almostembed into a topological space $X$, and $\mathcal F$ is a finite and intersectionclosed family of sets in $X$, such that each member of $\mathcal F$ has at most $b$ pathconnected components, then $r(\mathcal F)\leq p_G(b)$. In particular, if $\mathcal F$ is a family of sets in $\mathbb R^2$, $r(\mathcal F)\leq r_{K_{3,3}}(b)\leq 2b^6  5b^5 + 10b^4  10b^3 + 9b^2  3b + 3$. The proof further refines the constrained chain map method introduced by Goaoc, Paták, Patáková, Tancer and Wagner; and further developed by Patáková. In the plane, we manage to overcome the use of hypergraph Ramsey theorem and provide polynomial size bounds.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.08489
 Bibcode:
 2019arXiv190908489P
 Keywords:

 Mathematics  Combinatorics;
 52A35;
 68R10