Variations on Ramsey numbers and minimum numbers of monochromatic triangles in line $2$-colorings of configurations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Acceso en línea: | |
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| _version_ | 1866911854825046016 |
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| author | Bishop, Jamie Kuss, Rebekah Peet, Benjamin |
| author_facet | Bishop, Jamie Kuss, Rebekah Peet, Benjamin |
| contents | This paper begins by exploring some old and new results about Ramsey numbers and minimum numbers of monochromatic triangles in $2$-colorings of complete graphs, both in the disjoint and non-disjoint cases. We then extend the theory, by defining line $2$-colorings of configurations of points and lines and considering the minimum number of non-disjoint monochromatic triangles. We compute specific examples for notable symmetric $v_{3}$ configurations before considering a general result regarding the addition or connected sum of configurations through incidence switches. The paper finishes by considering the maximal number of mutually intersecting lines and how this relates to the minimum number of triangles given a line $2$-coloring of a symmetric $v_{3}$ configuration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_06912 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Variations on Ramsey numbers and minimum numbers of monochromatic triangles in line $2$-colorings of configurations Bishop, Jamie Kuss, Rebekah Peet, Benjamin Combinatorics 05C55, 51E30, 05B30 This paper begins by exploring some old and new results about Ramsey numbers and minimum numbers of monochromatic triangles in $2$-colorings of complete graphs, both in the disjoint and non-disjoint cases. We then extend the theory, by defining line $2$-colorings of configurations of points and lines and considering the minimum number of non-disjoint monochromatic triangles. We compute specific examples for notable symmetric $v_{3}$ configurations before considering a general result regarding the addition or connected sum of configurations through incidence switches. The paper finishes by considering the maximal number of mutually intersecting lines and how this relates to the minimum number of triangles given a line $2$-coloring of a symmetric $v_{3}$ configuration. |
| title | Variations on Ramsey numbers and minimum numbers of monochromatic triangles in line $2$-colorings of configurations |
| topic | Combinatorics 05C55, 51E30, 05B30 |
| url | https://arxiv.org/abs/2208.06912 |