Bipartite and Euclidean Gallai-Ramsey Theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929535488884736 |
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| author | McGuigan, Isabel Pan, Katherine |
| author_facet | McGuigan, Isabel Pan, Katherine |
| contents | In this paper, we investigate the following Gallai-Ramsey question: how large must a complete bipartite graph $K_{n_1, n_2}$ be before any coloring of its edges with $r$ colors contains either a monochromatic copy of $G = K_{s,t}$ or a rainbow copy of $H = K_{s,t}$? We demonstrate that the answer is linear in $r$, and provide more precise bounds for the specific case $s = 2$. Furthermore, we also consider the following Euclidean Gallai-Ramsey question: given a configuration $H$ in Euclidean space, what is the smallest $n$ such that any $r$-coloring of $n$-dimensional Euclidean space contains a monochromatic or rainbow configuration congruent to $H$? Through a natural translation between edge colorings of the complete bipartite graph $K_{n_1,n_2}$ and colorings of a subset of $(n_1+n_2)$-dimensional Euclidean space, we prove new upper bounds on $n$ for some configurations which can be expressed as Cartesian products of simplices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07634 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bipartite and Euclidean Gallai-Ramsey Theory McGuigan, Isabel Pan, Katherine Combinatorics In this paper, we investigate the following Gallai-Ramsey question: how large must a complete bipartite graph $K_{n_1, n_2}$ be before any coloring of its edges with $r$ colors contains either a monochromatic copy of $G = K_{s,t}$ or a rainbow copy of $H = K_{s,t}$? We demonstrate that the answer is linear in $r$, and provide more precise bounds for the specific case $s = 2$. Furthermore, we also consider the following Euclidean Gallai-Ramsey question: given a configuration $H$ in Euclidean space, what is the smallest $n$ such that any $r$-coloring of $n$-dimensional Euclidean space contains a monochromatic or rainbow configuration congruent to $H$? Through a natural translation between edge colorings of the complete bipartite graph $K_{n_1,n_2}$ and colorings of a subset of $(n_1+n_2)$-dimensional Euclidean space, we prove new upper bounds on $n$ for some configurations which can be expressed as Cartesian products of simplices. |
| title | Bipartite and Euclidean Gallai-Ramsey Theory |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.07634 |