On Bollobás-type theorems of $d$-tuples
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912174423670784 |
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| author | Yue, Erfei |
| author_facet | Yue, Erfei |
| contents | In 1965, Bollobás proved that for a Bollobás set-pair system $\{(A_i,B_i)\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i|+|B_i|}{A_i}^{-1}$ is $1$. Hegedüs and Frankl recently extended the concept of Bollobás systems to $d$-tuples, conjecturing that for a Bollobás system of $d$-tuples, $\{(A_i^{(1)},\ldots,A_i^{(d)})\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i^{(1)}|+\cdots+|A_i^{(d)}|}{|A_i^{(1)}|,\ldots,|A_i^{(d)}|}^{-1}$ is also $1$. This paper refutes this conjecture and establishes an upper bound for the sum. In the case $d=3$, the derived upper bound is asymptotically tight. Furthermore, we sharpen an inequality for skew Bollobás systems of $d$-tuples in Hegedüs and Frankl's paper. Finally, we determine the maximum size of a uniform skew Bollobás system of $d$-tuples on both sets and spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17192 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Bollobás-type theorems of $d$-tuples Yue, Erfei Combinatorics In 1965, Bollobás proved that for a Bollobás set-pair system $\{(A_i,B_i)\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i|+|B_i|}{A_i}^{-1}$ is $1$. Hegedüs and Frankl recently extended the concept of Bollobás systems to $d$-tuples, conjecturing that for a Bollobás system of $d$-tuples, $\{(A_i^{(1)},\ldots,A_i^{(d)})\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i^{(1)}|+\cdots+|A_i^{(d)}|}{|A_i^{(1)}|,\ldots,|A_i^{(d)}|}^{-1}$ is also $1$. This paper refutes this conjecture and establishes an upper bound for the sum. In the case $d=3$, the derived upper bound is asymptotically tight. Furthermore, we sharpen an inequality for skew Bollobás systems of $d$-tuples in Hegedüs and Frankl's paper. Finally, we determine the maximum size of a uniform skew Bollobás system of $d$-tuples on both sets and spaces. |
| title | On Bollobás-type theorems of $d$-tuples |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.17192 |