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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2407.05773 |
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| _version_ | 1866929413617090560 |
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| author | Girão, António Michel, Lukas Tamitegama, Youri |
| author_facet | Girão, António Michel, Lukas Tamitegama, Youri |
| contents | We say that a family of permutations $t$-shatters a set if it induces at least $t$ distinct permutations on that set. What is the minimum number $f_k(n,t)$ of permutations of $\{1, \dots, n\}$ that $t$-shatter all subsets of size $k$? For $t \le 2$, $f_k(n,t) = Θ(1)$. Spencer showed that $f_k(n,t) = Θ(\log \log n)$ for $3 \le t \le k$ and $f_k(n,k!) = Θ(\log n)$. In 1996, Füredi asked whether partial shattering with permutations must always fall into one of these three regimes. Johnson and Wickes recently settled the case $k = 3$ affirmatively and proved that $f_k(n,t) = Θ(\log n)$ for $t > 2 (k-1)!$.
We give a surprising negative answer to the question of Füredi by showing that a fourth regime exists for $k \ge 4$. We establish that $f_k(n,t) = Θ(\sqrt{\log n})$ for certain values of $t$ and prove that this is the only other regime when $k = 4$. We also show that $f_k(n,t) = Θ(\log n)$ for $t > 2^{k-1}$. This greatly narrows the range of $t$ for which the asymptotic behaviour of $f_k(n,t)$ is unknown. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05773 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small families of partially shattering permutations Girão, António Michel, Lukas Tamitegama, Youri Combinatorics We say that a family of permutations $t$-shatters a set if it induces at least $t$ distinct permutations on that set. What is the minimum number $f_k(n,t)$ of permutations of $\{1, \dots, n\}$ that $t$-shatter all subsets of size $k$? For $t \le 2$, $f_k(n,t) = Θ(1)$. Spencer showed that $f_k(n,t) = Θ(\log \log n)$ for $3 \le t \le k$ and $f_k(n,k!) = Θ(\log n)$. In 1996, Füredi asked whether partial shattering with permutations must always fall into one of these three regimes. Johnson and Wickes recently settled the case $k = 3$ affirmatively and proved that $f_k(n,t) = Θ(\log n)$ for $t > 2 (k-1)!$. We give a surprising negative answer to the question of Füredi by showing that a fourth regime exists for $k \ge 4$. We establish that $f_k(n,t) = Θ(\sqrt{\log n})$ for certain values of $t$ and prove that this is the only other regime when $k = 4$. We also show that $f_k(n,t) = Θ(\log n)$ for $t > 2^{k-1}$. This greatly narrows the range of $t$ for which the asymptotic behaviour of $f_k(n,t)$ is unknown. |
| title | Small families of partially shattering permutations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.05773 |