Intersection theorems for families of matchings of complete $k$-partite $k$-graphs
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866912670866735104 |
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| author | Mammoliti, Adam |
| author_facet | Mammoliti, Adam |
| contents | The celebrated {Erdős-Ko-Rado} Theorem states that for $n \geq 2k$ a family $\mathscr{F}$ of $k$ subsets of $[n]$ for which each pair of members of $\mathscr{F}$ have a non-empty intersection has size at most $\binom{n-1}{k-1}$ and for $n >2k$ has exactly this size if and only if it is the family of all $k$-subsets of $[n]$ containing a fixed element $x\in [n]$. Since its discovery, the {Erdős-Ko-Rado} Theorem has be generalised extensively and many variants have been found for structures other than sets. One such variant is for permutations and so-called generalised permutations. These structures are equivalent to $r$-matchings of the complete bipartite graph $K_{n,m}$ with $r \leq \min\{n,m\}$ in a natural way.
The culmination of results of several groups of authors constitute an {Erdős-Ko-Rado} Theorem for families of generalised permutations and so for families of $r$-matchings of $K_{n,m}$ for all feasible values of $r,n$ and $m$. In this paper we generalise this by proving an {Erdős-Ko-Rado} Theorem for families of $r$-matchings of complete $k$-partite $k$-graphs, which can be seen as a partial generalisation of the {Erdős-Ko-Rado} Theorem itself. We also prove similar results for $t$-intersecting families, and for families of matchings whose members have sizes from some set of integers $R$, rather than a single size $r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_07481 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Intersection theorems for families of matchings of complete $k$-partite $k$-graphs Mammoliti, Adam Combinatorics 05D05, 05C65, 05A05 The celebrated {Erdős-Ko-Rado} Theorem states that for $n \geq 2k$ a family $\mathscr{F}$ of $k$ subsets of $[n]$ for which each pair of members of $\mathscr{F}$ have a non-empty intersection has size at most $\binom{n-1}{k-1}$ and for $n >2k$ has exactly this size if and only if it is the family of all $k$-subsets of $[n]$ containing a fixed element $x\in [n]$. Since its discovery, the {Erdős-Ko-Rado} Theorem has be generalised extensively and many variants have been found for structures other than sets. One such variant is for permutations and so-called generalised permutations. These structures are equivalent to $r$-matchings of the complete bipartite graph $K_{n,m}$ with $r \leq \min\{n,m\}$ in a natural way. The culmination of results of several groups of authors constitute an {Erdős-Ko-Rado} Theorem for families of generalised permutations and so for families of $r$-matchings of $K_{n,m}$ for all feasible values of $r,n$ and $m$. In this paper we generalise this by proving an {Erdős-Ko-Rado} Theorem for families of $r$-matchings of complete $k$-partite $k$-graphs, which can be seen as a partial generalisation of the {Erdős-Ko-Rado} Theorem itself. We also prove similar results for $t$-intersecting families, and for families of matchings whose members have sizes from some set of integers $R$, rather than a single size $r$. |
| title | Intersection theorems for families of matchings of complete $k$-partite $k$-graphs |
| topic | Combinatorics 05D05, 05C65, 05A05 |
| url | https://arxiv.org/abs/1811.07481 |