The maximum sum of the size of all intersections within intersecting families and crossing-intersecting families
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913244409495552 |
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| author | Huang, Sumin |
| author_facet | Huang, Sumin |
| contents | Let $ω(\mathcal{F})=\sum_{\{A,B\}\subset\mathcal{F}}|A\cap B|$ and $ω(\mathcal{A},\mathcal{B})=\sum_{(A,B)\in \mathcal{A}\times \mathcal{B}}|A\cap B|$. A family $\mathcal{F}$ is intersecting if $F_1\cap F_2\neq \emptyset$ for any $F_1,F_2\in\mathcal{F}$ and two family $\mathcal{A}$ and $\mathcal{B}$ are crossing-intersecting if $A\cap B\neq \emptyset$ for any $(A,B)\in \mathcal{A}\times\mathcal{B}$. For an intersecting family $\mathcal{F}$, Erdős, Ko and Rado determined the upper bound of $|\mathcal{F}|$, consequently yielding an upper bound of $\binom{|\mathcal{F}|}{2}=\sum_{\{A,B\}\subset\mathcal{F}}1$. If we replace $1$ with $|A\cap B|$ in the summation $\sum_{\{A,B\}\subset\mathcal{F}}1$, then this summation transforms into $ω(\mathcal{F})$. In this paper, for an intersecting family $\mathcal{F}$, we determine the upper bound of $ω(\mathcal{F})$, which is a generalization of Erdős-Ko-Rado Theorem. Further, for crossing-intersecting families $\mathcal{A}$ and $\mathcal{B}$, we determine the upper bound of $ω(\mathcal{A},\mathcal{B})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16730 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The maximum sum of the size of all intersections within intersecting families and crossing-intersecting families Huang, Sumin Combinatorics Let $ω(\mathcal{F})=\sum_{\{A,B\}\subset\mathcal{F}}|A\cap B|$ and $ω(\mathcal{A},\mathcal{B})=\sum_{(A,B)\in \mathcal{A}\times \mathcal{B}}|A\cap B|$. A family $\mathcal{F}$ is intersecting if $F_1\cap F_2\neq \emptyset$ for any $F_1,F_2\in\mathcal{F}$ and two family $\mathcal{A}$ and $\mathcal{B}$ are crossing-intersecting if $A\cap B\neq \emptyset$ for any $(A,B)\in \mathcal{A}\times\mathcal{B}$. For an intersecting family $\mathcal{F}$, Erdős, Ko and Rado determined the upper bound of $|\mathcal{F}|$, consequently yielding an upper bound of $\binom{|\mathcal{F}|}{2}=\sum_{\{A,B\}\subset\mathcal{F}}1$. If we replace $1$ with $|A\cap B|$ in the summation $\sum_{\{A,B\}\subset\mathcal{F}}1$, then this summation transforms into $ω(\mathcal{F})$. In this paper, for an intersecting family $\mathcal{F}$, we determine the upper bound of $ω(\mathcal{F})$, which is a generalization of Erdős-Ko-Rado Theorem. Further, for crossing-intersecting families $\mathcal{A}$ and $\mathcal{B}$, we determine the upper bound of $ω(\mathcal{A},\mathcal{B})$. |
| title | The maximum sum of the size of all intersections within intersecting families and crossing-intersecting families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.16730 |