On<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>r</mml:mi></mml:math>-cross<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" display="inline" overflow="scroll"><mml:mi>t</mml:mi></mml:math>-intersecting families for weak compositions

Ku, Cheng Yeaw; Wong, Kok Bin

doi:10.1016/j.disc.2015.01.015

Cited by 5 publications

(3 citation statements)

References 27 publications

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“…Ku and Wong [41] solved the t-intersection problem for C n,r with n sufficiently large. In [42], they also proved Theorem 4.10 below for sufficiently large values of n 1 , . .…”

Section: Families Of Compositionsmentioning

confidence: 87%

The maximum product of sizes of cross-intersecting families

Borg

2017

Discrete Mathematics

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We say that a set A t-intersects a set B if A and B have at least t common elements. Two families A and B of sets are said to be cross-t-intersecting if each set in A t-intersects each set in B. A subfamily S of a family F is called a t-star of F if the sets in S have t common elements. Let l(F, t) denote the size of a largest t-star of F. We call F a (≤ r)-family if each set in F has at most r elements. We determine a function c : N 3 → N such that the following holds. If A is a subfamily of a (≤ r)-family F with l(F, t) ≥ c(r, s, t)l(F, t + 1), B is a subfamily of a (≤ s)-family G with l(G, t) ≥ c(r, s, t)l(G, t + 1), and A and B are cross-t-intersecting, then |A||B| ≤ l(F, t)l(G, t). Some known results follow from this, and we identify several natural classes of families for which the bound is attained.

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“…Ku and Wong [41] solved the t-intersection problem for C n,r with n sufficiently large. In [42], they also proved Theorem 4.10 below for sufficiently large values of n 1 , . .…”

Section: Families Of Compositionsmentioning

confidence: 87%

The maximum product of sizes of cross-intersecting families

Borg

2017

Discrete Mathematics

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“…In the celebrated paper [1], Ahlswede and Khachatrian extended the Erdős-Ko-Rado theorem by determining the structure of all t-intersecting set systems of maximum size for all possible n (see also [3,14,16,17,20,25,31,36,40,42,43,45] for some related results). There have been many recent results showing that a version of the Erdős-Ko-Rado theorem holds for combinatorial objects other than set systems.…”

Section: A|mentioning

confidence: 99%

“…The investigation of the Erdős-Ko-Rado property for graphs started in [23], and gave rise to [4,6,21,22,24,47]. The Erdős-Ko-Rado type results also appear in vector spaces [9,18], set partitions [27,29,30] and weak compositions [32,33,34].…”

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

Wong

2016

Discrete Mathematics

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Onr-crosst-intersecting families for weak compositions

Cited by 5 publications

References 27 publications

The maximum product of sizes of cross-intersecting families

The maximum product of sizes of cross-intersecting families

A Deza–Frankl Type Theorem for Set Partitions

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

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