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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2212.05353 |
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| _version_ | 1866917915478982656 |
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| author | Crager, Julia Flores, Felicia Goldberg, Timothy E. Rose, Lauren L. Rose-Levine, Daniel Thornburgh, Darrion Walker, Raphael |
| author_facet | Crager, Julia Flores, Felicia Goldberg, Timothy E. Rose, Lauren L. Rose-Levine, Daniel Thornburgh, Darrion Walker, Raphael |
| contents | We define a \textit{cap} in the affine geometry $AG(n,2)$ to be a subset in which any collection of 4 points is in general position. In this paper we classify, up to affine equivalence, all caps in $AG(n,2)$ of size $k \leq 9$. As a result, we obtain a complete characterization of caps in dimension $n \leq 6$, in particular complete and maximal caps. Since the \textit{EvenQuads} card deck is a model for $AG(6,2)$, as a consequence we determine the probability that an arbitrary $k$-card layout contains a quad. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_05353 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | How many cards should you lay out in a game of EvenQuads?: A detailed study of caps in AG(n,2) Crager, Julia Flores, Felicia Goldberg, Timothy E. Rose, Lauren L. Rose-Levine, Daniel Thornburgh, Darrion Walker, Raphael Combinatorics 05B25, 51E10, 51E15 We define a \textit{cap} in the affine geometry $AG(n,2)$ to be a subset in which any collection of 4 points is in general position. In this paper we classify, up to affine equivalence, all caps in $AG(n,2)$ of size $k \leq 9$. As a result, we obtain a complete characterization of caps in dimension $n \leq 6$, in particular complete and maximal caps. Since the \textit{EvenQuads} card deck is a model for $AG(6,2)$, as a consequence we determine the probability that an arbitrary $k$-card layout contains a quad. |
| title | How many cards should you lay out in a game of EvenQuads?: A detailed study of caps in AG(n,2) |
| topic | Combinatorics 05B25, 51E10, 51E15 |
| url | https://arxiv.org/abs/2212.05353 |