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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.05674 |
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| _version_ | 1866910494911102976 |
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| author | Grill, Karl Linzmayer, Daniel Wien, TU |
| author_facet | Grill, Karl Linzmayer, Daniel Wien, TU |
| contents | If an $n$-uniform hypergraph can be 2-colored, then it is said to have property B. Erdős (1963) was the first to give lower and upper bounds for the minimal size $m(n)$ of an $n$-uniform hypergraph without property B. His asymptotic upper bound $O(n^22^n)$ still is the best we know, his lower bound $2^{n-1}$ has seen a number of improvements, with the current best $Ω$ $(2^n\sqrt{n/\log(n)})$ established by Radhakrishnan and Srinivasan (2000). Cherkashin and Kozik (2014) provided a simplified proof of this result, using Pluhár's (2009) idea of a random greedy coloring. In the present paper, we use a refined version of this argument to obtain improved lower bounds on $m(n)$ for small values of $n$. We also study $m(n,v)$, the size of the smallest $n$-hypergraph without property B having $v$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05674 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Lower Bounds for Property B Grill, Karl Linzmayer, Daniel Wien, TU Combinatorics Probability 05C30 (Primary) 05C65 (Secondary) If an $n$-uniform hypergraph can be 2-colored, then it is said to have property B. Erdős (1963) was the first to give lower and upper bounds for the minimal size $m(n)$ of an $n$-uniform hypergraph without property B. His asymptotic upper bound $O(n^22^n)$ still is the best we know, his lower bound $2^{n-1}$ has seen a number of improvements, with the current best $Ω$ $(2^n\sqrt{n/\log(n)})$ established by Radhakrishnan and Srinivasan (2000). Cherkashin and Kozik (2014) provided a simplified proof of this result, using Pluhár's (2009) idea of a random greedy coloring. In the present paper, we use a refined version of this argument to obtain improved lower bounds on $m(n)$ for small values of $n$. We also study $m(n,v)$, the size of the smallest $n$-hypergraph without property B having $v$ vertices. |
| title | Improved Lower Bounds for Property B |
| topic | Combinatorics Probability 05C30 (Primary) 05C65 (Secondary) |
| url | https://arxiv.org/abs/2403.05674 |