Saturation Problems in Convex Geometric Hypergraphs

TL;DR

This paper investigates saturation problems in convex geometric hypergraphs, providing asymptotic results for specific configurations and determining saturation numbers for most small cases.

Contribution

It offers the first asymptotic determination of saturation numbers for certain convex geometric hypergraph configurations and analyzes multiple small cases.

Findings

Asymptotically determined saturation number for two disjoint r-tuples.

Saturation numbers for seven out of eight 3-uniform cgh's on two edges.

Approximate saturation number for the eighth case within a log factor.

Abstract

A convex geometric hypergraph (abbreviated cgh) consists of a collection of subsets of a strictly convex set of points in the plane. Extremal problems for cgh's have been extensively studied in the literature, and in this paper we consider their corresponding saturation problems. We asymptotically determine the saturation number of two geometrically disjoint $r$-tuples. Further, amongst the eight nonisomorphic $3$-uniform cgh's on two edges, we determine the saturation number for seven of these up to order of magnitude and the eighth up to a log factor.