Convex geometries representable by at most 5 circles on the plane

TL;DR

This paper classifies convex geometries on small sets based on their representability by circles on the plane, showing most can be represented except for a few with specific properties.

Contribution

It provides a complete classification of convex geometries on 4- and 5-element sets regarding circle representations, identifying new non-representable cases.

Findings

All 34 geometries on 4-element sets are representable by circles.

Out of 672 geometries on 5-element sets, 623 are representable.

7 geometries on 5-element sets are proven non-representable due to the Triangle Property.

Abstract

A convex geometry is a closure system satisfying the anti-exchange property. In this work we document all convex geometries on 4- and 5-element base sets with respect to their representation by circles on the plane. All 34 non-isomorphic geometries on a 4-element set can be represented by circles, and of the 672 geometries on a 5-element set, we made representations of 623. Of the 49 remaining geometries on a 5-element set, one was already shown not to be representable due to the Weak Carousel property, as articulated by Adaricheva and Bolat (Discrete Mathematics, 2019). In this paper we show that 7 more of these convex geometries cannot be represented by circles on the plane, due to what we term the Triangle Property.