Convex geometries representable by at most 5 circles on the plane
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arXiv
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| Format: | Preprint |
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2020
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| author | PolyMath REU Convex Geometries Collaboration Adaricheva, Kira Bolat, Madina Gjonbalaj, Gent Amerine, Brandon Behne, J. Alexandria Daisy, Evan Frederiksen, Alexander Garg, Ayush King, Zachary Ma, Grace Olson, Michelle Pai, Rohit Park, Junewoo Raanes, Cat Riedel, Sean Rogge, Joseph Sarch, Raviv Thompson, James Yepez-Lopez, Fernanda Zhou, Stephanie |
| author_facet | PolyMath REU Convex Geometries Collaboration Adaricheva, Kira Bolat, Madina Gjonbalaj, Gent Amerine, Brandon Behne, J. Alexandria Daisy, Evan Frederiksen, Alexander Garg, Ayush King, Zachary Ma, Grace Olson, Michelle Pai, Rohit Park, Junewoo Raanes, Cat Riedel, Sean Rogge, Joseph Sarch, Raviv Thompson, James Yepez-Lopez, Fernanda Zhou, Stephanie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_13077 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Convex geometries representable by at most 5 circles on the plane PolyMath REU Convex Geometries Collaboration Adaricheva, Kira Bolat, Madina Gjonbalaj, Gent Amerine, Brandon Behne, J. Alexandria Daisy, Evan Frederiksen, Alexander Garg, Ayush King, Zachary Ma, Grace Olson, Michelle Pai, Rohit Park, Junewoo Raanes, Cat Riedel, Sean Rogge, Joseph Sarch, Raviv Thompson, James Yepez-Lopez, Fernanda Zhou, Stephanie Combinatorics 05B25, 51D20, 52C05, 52-11 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. |
| title | Convex geometries representable by at most 5 circles on the plane |
| topic | Combinatorics 05B25, 51D20, 52C05, 52-11 |
| url | https://arxiv.org/abs/2008.13077 |