On the Erdős-Ko-Rado problem of flags with type $\{1, n-3 \}$ of finite sets
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| Format: | Preprint |
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2025
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| _version_ | 1866916810913218560 |
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| author | Heering, Philipp |
| author_facet | Heering, Philipp |
| contents | A flag of a finite set $S$ is a set $f$ of non-empty, proper subsets of $S$, such that $X\subseteq Y$ or $Y\subseteq X$ for all $X,Y\in f$. Two flags $f_1$ and $f_2$ of $S$ are opposite if $X_1\cap X_2=\emptyset$, or $X_1\cup X_2=S$ for all $X_1\in f_1$ and $X_2\in f_2$. The set $\{|X| \mid X\in f \}$ is the type of a flag $f$. A set of pairwise non-opposite flags is an Erdős-Ko-Rado set. In 2022 Metsch posed the problem of determining the maximum size of all Erdős-Ko-Rado sets of flags of type $T$ with $|T|=2$. We contribute towards this by determining the maximum size for flags of type $\{ 1,n-3\}$ for finite sets with $n$ elements. Furthermore we answer an open questions of Metsch regarding a small case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Erdős-Ko-Rado problem of flags with type $\{1, n-3 \}$ of finite sets Heering, Philipp Combinatorics 05C69, 05D05, 05C35 A flag of a finite set $S$ is a set $f$ of non-empty, proper subsets of $S$, such that $X\subseteq Y$ or $Y\subseteq X$ for all $X,Y\in f$. Two flags $f_1$ and $f_2$ of $S$ are opposite if $X_1\cap X_2=\emptyset$, or $X_1\cup X_2=S$ for all $X_1\in f_1$ and $X_2\in f_2$. The set $\{|X| \mid X\in f \}$ is the type of a flag $f$. A set of pairwise non-opposite flags is an Erdős-Ko-Rado set. In 2022 Metsch posed the problem of determining the maximum size of all Erdős-Ko-Rado sets of flags of type $T$ with $|T|=2$. We contribute towards this by determining the maximum size for flags of type $\{ 1,n-3\}$ for finite sets with $n$ elements. Furthermore we answer an open questions of Metsch regarding a small case. |
| title | On the Erdős-Ko-Rado problem of flags with type $\{1, n-3 \}$ of finite sets |
| topic | Combinatorics 05C69, 05D05, 05C35 |
| url | https://arxiv.org/abs/2506.20556 |