A non-trivial intersection theorem for permutations with fixed number of cycles

Ku, Cheng Yeaw; Wong, Kok Bin

doi:10.1016/j.disc.2015.09.032

Cited by 3 publications

(2 citation statements)

References 44 publications

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“…Intersecting families of permutations were initiated by Deza and Frankl in [10]. Some recent work done on this problem and its variants can be found in [5,7,8,11,12,19,26,28,35,37,38,39,44]. The investigation of the Erdős-Ko-Rado property for graphs started in [23], and gave rise to [4,6,21,22,24,47].…”

Section: A|mentioning

confidence: 99%

A Deza–Frankl Type Theorem for Set Partitions

Wong

2015

Electron. J. Combin.

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A set partition of $[n]$ is a collection of pairwise disjoint nonempty subsets (called blocks) of $[n]$ whose union is $[n]$. Let $\mathcal{B}(n)$ denote the family of all set partitions of $[n]$. A family $\mathcal{A} \subseteq \mathcal{B}(n)$ is said to be $m$-intersecting if any two of its members have at least $m$ blocks in common. For any set partition $P \in \mathcal{B}(n)$, let $\tau(P) = \{x: \{x\} \in P\}$ denote the union of its singletons. Also, let $\mu(P) = [n] -\tau(P)$ denote the set of elements that do not appear as a singleton in $P$. Let \begin{align*} {\mathcal F}_{2t} & =\left\{P \in \mathcal{B}(n)\ : \ \vert \mu (P)\vert\leq t\right\};\\{\mathcal F}_{2t+1}(i_0) & =\left\{P \in \mathcal{B}(n)\ : \ \vert\mu (P)\cap ([n]\setminus \{i_0\})\vert\leq t\right\}.\end{align*} In this paper, we show that for $r\geq 3$, there exists a $n_0=n_0(r)$ depending on $r$ such that for all $n\geq n_0$, if $\mathcal{A} \subseteq\mathcal{B}(n)$ is $(n-r)$-intersecting, then \[ |\mathcal{A}| \leq \begin{cases} \vert {\mathcal F}_{2t} \vert, & \text{if $r=2t$};\\ \vert {\mathcal F}_{2t+1}(1) \vert, & \text{if $r=2t+1$}.\end{cases}\]Moreover, equality holds if and only if \[ \mathcal{A}= \begin{cases} {\mathcal F}_{2t}, & \text{if $r=2t$};\\ {\mathcal F}_{2t+1}(i_0), & \text{if $r=2t+1$},\end{cases}\]for some $i_0\in [n]$.

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Section: A|mentioning

confidence: 99%

A Deza–Frankl Type Theorem for Set Partitions

Wong

2015

Electron. J. Combin.

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“…In recent years, considering the intersecting families of the subgroups of S n and cross-intersecting family in other mathematical objects have result in much attention. See [10][11][12][13][14] for versions of intersecting family for A n , GL(n, q), PGL(2, q), PSL(2, q) and so on. See [15][16][17] for a version of a cross-intersecting family.…”

Section: Introductionmentioning

confidence: 99%

The Cross-Intersecting Family of Certain Permutation Groups

Han

2023

Symmetry

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Two subsets X and Y of a permutation group G acting on Ω are cross-intersecting if for every x∈X and every y∈Y there exists some point α∈Ω such that αx=αy. Based on several observations made on the cross-independent version of Hoffman’s theorem, we characterize in this paper the cross-intersecting families of certain permutation groups. Our proof uses a Cayley graph on a permutation subgroup with respect to the derangement. By carefully analyzing the cross-independent version of Hoffman’s theorem, we obtain a useful theorem to consider cross-intersecting subsets of certain kinds of permutation subgroups, such as PGL(2,q), PSL(2,q) and Sn.

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Erdős-Ko-Rado theorems for set partitions with certain block size

Wong

2020

Journal of Combinatorial Theory, Series A

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A non-trivial intersection theorem for permutations with fixed number of cycles

Cited by 3 publications

References 44 publications

A Deza–Frankl Type Theorem for Set Partitions

A Deza–Frankl Type Theorem for Set Partitions

The Cross-Intersecting Family of Certain Permutation Groups

Erdős-Ko-Rado theorems for set partitions with certain block size

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