Exact results on traces of sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929401691635712 |
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| author | Li, Mingze Ma, Jie Rong, Mingyuan |
| author_facet | Li, Mingze Ma, Jie Rong, Mingyuan |
| contents | For non-negative integers $n$, $m$, $a$ and $b$, we write $\left( n,m \right) \rightarrow \left( a,b \right)$ if for every family $\mathcal{F}\subseteq 2^{[n]}$ with $|\mathcal{F}|\geqslant m$ there is an $a$-element set $T\subseteq [n]$ such that $\left| \mathcal{F}_{\mid T} \right| \geqslant b$, where $\mathcal{F}_{\mid T}=\{ F \cap T : F \in \mathcal{F} \}$. A longstanding problem in extremal set theory asks to determine $m(s)=\lim_{n\rightarrow +\infty}\frac{m(n,s)}{n}$, where $m(n,s)$ denotes the maximum integer $m$ such that $\left( n,m \right) \rightarrow \left( n-1,m-s \right)$ holds for non-negatives $n$ and $s$. In this paper, we establish the exact value of $m(2^{d-1}-c)$ for all $1\leqslant c\leqslant d$ whenever $d\geqslant 50$, thereby solving an open problem posed by Piga and Schülke. To be precise, we show that $$m(n,2^{d-1}-c)=\frac{2^{d}-c}{d}n \mbox{ for } 1\leq c\leq d-1 \mbox{ and } d\mid n, \mbox{ and }
m(n,2^{d-1}-d)=\frac{2^{d}-d-0.5}{d}n \mbox{ for } 2d\mid n $$ holds for $d\geq 50$. Furthermore, we provide a proof that confirms a conjecture of Frankl and Watanabe from 1994, demonstrating that $m(11)=5.3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact results on traces of sets Li, Mingze Ma, Jie Rong, Mingyuan Combinatorics For non-negative integers $n$, $m$, $a$ and $b$, we write $\left( n,m \right) \rightarrow \left( a,b \right)$ if for every family $\mathcal{F}\subseteq 2^{[n]}$ with $|\mathcal{F}|\geqslant m$ there is an $a$-element set $T\subseteq [n]$ such that $\left| \mathcal{F}_{\mid T} \right| \geqslant b$, where $\mathcal{F}_{\mid T}=\{ F \cap T : F \in \mathcal{F} \}$. A longstanding problem in extremal set theory asks to determine $m(s)=\lim_{n\rightarrow +\infty}\frac{m(n,s)}{n}$, where $m(n,s)$ denotes the maximum integer $m$ such that $\left( n,m \right) \rightarrow \left( n-1,m-s \right)$ holds for non-negatives $n$ and $s$. In this paper, we establish the exact value of $m(2^{d-1}-c)$ for all $1\leqslant c\leqslant d$ whenever $d\geqslant 50$, thereby solving an open problem posed by Piga and Schülke. To be precise, we show that $$m(n,2^{d-1}-c)=\frac{2^{d}-c}{d}n \mbox{ for } 1\leq c\leq d-1 \mbox{ and } d\mid n, \mbox{ and } m(n,2^{d-1}-d)=\frac{2^{d}-d-0.5}{d}n \mbox{ for } 2d\mid n $$ holds for $d\geq 50$. Furthermore, we provide a proof that confirms a conjecture of Frankl and Watanabe from 1994, demonstrating that $m(11)=5.3$. |
| title | Exact results on traces of sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.18870 |