Size, diversity, minimum degree, sturdiness, dömdödöm
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913643965186048 |
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| author | Patkós, Balázs |
| author_facet | Patkós, Balázs |
| contents | For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}: A\subseteq F, ~B\cap F=\emptyset\}$. The \textbf{$(p,q)$-dömdödöm} of a family $\mathcal{F}\subseteq 2^{[n]}$ is $β_{p,q}(\mathcal{F})=\min\{|\mathcal{F}(A,\overline{B})|:|A|=p,|B|=q, A\cap B=\emptyset, A,B\subseteq [n]\} $. This definition encompasses size, diversity, minimum degree, and sturdiness as special cases. We investigate the maximum possible value $β_{p,q}(n,k)$ of $β_{p,q}(\mathcal{F})$ over all $k$-uniform intersecting families $\mathcal{F}\subset 2^{[n]}$. We determine the order of magnitude of $β_{p,q}(n,k)$ for all fixed $p,q,k$. We relate the asymptotics of $β_{p,q}(n,k)$ to the constant value of $β_{0,q}(n,q+1)$ and establish $β_{p,1}(n,k)=\binom{n-3-p}{k-2-p}$ and $β_{p,2}(n,k)=2\binom{n-5}{k-3-p}-\binom{n-7}{k-5-p}$ if $n$ is large enough. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_02596 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Size, diversity, minimum degree, sturdiness, dömdödöm Patkós, Balázs Combinatorics For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}: A\subseteq F, ~B\cap F=\emptyset\}$. The \textbf{$(p,q)$-dömdödöm} of a family $\mathcal{F}\subseteq 2^{[n]}$ is $β_{p,q}(\mathcal{F})=\min\{|\mathcal{F}(A,\overline{B})|:|A|=p,|B|=q, A\cap B=\emptyset, A,B\subseteq [n]\} $. This definition encompasses size, diversity, minimum degree, and sturdiness as special cases. We investigate the maximum possible value $β_{p,q}(n,k)$ of $β_{p,q}(\mathcal{F})$ over all $k$-uniform intersecting families $\mathcal{F}\subset 2^{[n]}$. We determine the order of magnitude of $β_{p,q}(n,k)$ for all fixed $p,q,k$. We relate the asymptotics of $β_{p,q}(n,k)$ to the constant value of $β_{0,q}(n,q+1)$ and establish $β_{p,1}(n,k)=\binom{n-3-p}{k-2-p}$ and $β_{p,2}(n,k)=2\binom{n-5}{k-3-p}-\binom{n-7}{k-5-p}$ if $n$ is large enough. |
| title | Size, diversity, minimum degree, sturdiness, dömdödöm |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.02596 |