Erdős–Ko–Rado theorems for simplicial complexes

Woodroofe, Russ

doi:10.1016/j.jcta.2010.11.022

Cited by 22 publications

(13 citation statements)

References 31 publications

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“…What is the value of η(r, s, t)? A graph G is a pair (V, E) with E ⊆ V 2 , and a subset S of V is called an independent set of G if {i, j} / ∈ E for every i, j ∈ S. Let I G denote the family of all independent sets of a graph G. The EKR problem for I G was introduced in [28] and inspired many results [10,11,27,28,29,43]. Many EKR-type results can be phrased in terms of independent sets of graphs; see [11, page 2878].…”

Section: The Main Resultsmentioning

confidence: 99%

Cross-intersecting non-empty uniform subfamilies of hereditary families

Borg

2019

European Journal of Combinatorics

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A set A t-intersects a set B if A and B have at least t common elements. A set of sets is called a family. Two families A and B are cross-the size of a smallest base of H. We show that for any integers t, r, and s with 1 ≤ t ≤ r ≤ s, there exists an integer c(r, s, t) such that the following holds for any hereditary family H with µ(H) ≥ c(r, s, t). If A is a non-empty subfamily of H (r) , B is a non-empty subfamily of H (s) , A and B are cross-t-intersecting, and |A| + |B| is maximum under the given conditions, then for some set I in H with t ≤ |I| ≤ r, either A = {A ∈ H (r) : I ⊆ A} and B = {B ∈ H (s) : |B ∩ I| ≥ t}, or r = s, t < |I|, A = {A ∈ H (r) : |A ∩ I| ≥ t}, and B = {B ∈ H (s) : I ⊆ B}. This was conjectured by the author for t = 1 and generalizes well-known results for the case where H is a power set.

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Section: The Main Resultsmentioning

confidence: 99%

Cross-intersecting non-empty uniform subfamilies of hereditary families

Borg

2019

European Journal of Combinatorics

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“…They conjectured that I G (r) has the 1-star property if µ(I G ) ≥ 2r [31,Conjecture 7]. This conjecture inspired many results; see, for example, [30,31,13,32,47]. Clearly, I G is a hereditary family, so Theorem 1.3 verifies the conjecture for µ(I G ) ≥ 3 2 (r − 1) 2 (3r − 4) + r. Kamat [34] made the following analogous conjecture for cross-intersecting families.…”

Section: Moreover Equality Holds If and Only If One Of The Followingmentioning

confidence: 92%

Cross-intersecting subfamilies of levels of hereditary families

Borg

2020

Discrete Mathematics

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A set A t-intersects a set B if A and B have at least t common elements. Families A 1 , A 2 , . . . , A k of sets are cross-t-intersecting if, for every i and j in {1, 2, . . . , k} with i = j, each set in A i t-intersects each set in A j . An active problem in extremal set theory is to determine, for a given finite family F, the structure of k cross-t-intersecting subfamilies whose sum or product of sizes is maximum. For a family H, the r-th level H (r) of H is the family of all sets in H of size r, and, for s ≤ r, H (s) is called a (≤ r)-level of H. We solve the problem for any union F of (≤ r)-levels of any union H of power sets of sets of size at least a certain integer n 0 , where n 0 is independent of H and k but depends on r and t (dependence on r is inevitable, but dependence on t can be avoided). Our primary result asserts that there are only two possible optimal configurations for the sum. A special case was conjectured by Kamat in 2011. We also prove generalizations, whereby A 1 , A 2 , . . . , A k are not necessarily contained in the same union of levels. Various Erdős-Ko-Rado-type results follow. The sum problem for a level of a power set was solved for t = 1 by Hilton in 1977, and for any t by Wang and Zhang in 2011.

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“…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]. 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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Erdős–Ko–Rado theorems for simplicial complexes

Cited by 22 publications

References 31 publications

Cross-intersecting non-empty uniform subfamilies of hereditary families

Cross-intersecting non-empty uniform subfamilies of hereditary families

Cross-intersecting subfamilies of levels of hereditary families

A Deza–Frankl Type Theorem for Set Partitions

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