On fractional Helly properties in graphs without large complete minors

Abstract

Helly's theorem is a classical result about intersection patterns of convex sets in Euclidean spaces. It states that every finite family of convex sets in $R^d$ is intersecting if every d+1 or fewer members are intersecting. There are a large number of generalizations and applications of Helly's theorem. For instance, the colorful and fractional Helly theorems and the (p,q)-theorem are important generalizations. It was proved by Alon et al. that the fractional Helly property is a key ingredient when establishing the (p,q)-theorem for abstract set-systems. Hence it has been one of the most fundamental questions in this field to find a sufficient condition that makes a non-empty family of sets satisfies the fractional Helly property. In this dissertation, we give an overview of classical Helly type theorems, and prove some combinatorial generalizations of Helly's theorem with a focus on the fractional Helly properties.

Our first main topic is a combination of the colorful Helly theorem and the fractional Helly theorem. We introduce a purely combinatorial method to improve the bound of $B\'{a}r\'{a}ny$ et al.'s colorful fractional Helly theorem for convex sets. This proof technique can be easily modified to achieve a colorful fractional Helly theorem for abstract set-systems. The second main topic is about Helly type theorems for certain families of graphs. A connected cover in a graph G is a non-empty family $\mathcal{G}$ of induced subgraphs in G such that every non-empty intersection is connected. We prove some Helly type theorems for connected covers in graphs without large complete minors. In particular, the results on connected covers in planar graphs can be applied for set-systems which are more general than good covers in the plane.