scholarly journals Intersecting $k$-Uniform Families Containing all the $k$-Subsets of a Given Set

10.37236/3213 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Wei-Tian Li ◽  
Bor-Liang Chen ◽  
Kuo-Ching Huang ◽  
Ko-Wei Lih
Keyword(s):  
Nonempty Intersection ◽  
Intersecting Family ◽  
Intersecting Families

Let $m, n$, and $k$ be integers satisfying $0 < k \leq n < 2k \leq m$. A family of sets $\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\binom{[n]}{k} \subseteq \mathcal{F} \subseteq \binom{[m]}{k}$ and any pair of members of $\mathcal{F}$ have nonempty intersection. Maximum $(m,k,k)$- and $(m,k+1,k)$-intersecting families are determined by the theorems of Erdős-Ko-Rado and Hilton-Milner, respectively. We determine the maximum families for the cases $n = 2k-1, 2k-2, 2k-3$, and $m$ sufficiently large.

scholarly journals An Erdős-Ko-Rado Type Theorem via the Polynomial Method

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 640
Author(s):  
Kyung-Won Hwang ◽  
Younjin Kim ◽  
Naeem N. Sheikh
Keyword(s):  
Upper Bound ◽  
Type Theorem ◽  
Nonempty Intersection ◽  
Polynomial Method ◽  
Linearly Independent ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Families Of Subsets

A family F is an intersecting family if any two members have a nonempty intersection. Erdős, Ko, and Rado showed that | F | ≤ n − 1 k − 1 holds for a k-uniform intersecting family F of subsets of [ n ] . The Erdős-Ko-Rado theorem for non-uniform intersecting families of subsets of [ n ] of size at most k can be easily proved by applying the above result to each uniform subfamily of a given family. It establishes that | F | ≤ n − 1 k − 1 + n − 1 k − 2 + ⋯ + n − 1 0 holds for non-uniform intersecting families of subsets of [ n ] of size at most k. In this paper, we prove that the same upper bound of the Erdős-Ko-Rado Theorem for k-uniform intersecting families of subsets of [ n ] holds also in the non-uniform family of subsets of [ n ] of size at least k and at most n − k with one more additional intersection condition. Our proof is based on the method of linearly independent polynomials.

scholarly journals Counting Intersecting and Pairs of Cross-Intersecting Families

2017 ◽  
Vol 27 (1) ◽  
pp. 60-68 ◽  
Author(s):  
PETER FRANKL ◽  
ANDREY KUPAVSKII
Keyword(s):  
Classical Result ◽  
Maximum Size ◽  
Extremal Combinatorics ◽  
Intersecting Family ◽  
Intersecting Families

A family of subsets of {1,. . .,n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko and Rado determines the maximum size of an intersecting family of k-subsets of {1,. . .,n}. In this paper we study the following problem: How many intersecting families of k-subsets of {1,. . .,n} are there? Improving a result of Balogh, Das, Delcourt, Liu and Sharifzadeh, we determine this quantity asymptotically for n ≥ 2k+2+2$\sqrt{k\log k}$ and k → ∞. Moreover, under the same assumptions we also determine asymptotically the number of non-trivial intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.

scholarly journals Fractional L-intersecting Families

10.37236/7846 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Niranjan Balachandran ◽  
Rogers Mathew ◽  
Tapas Kumar Mishra
Keyword(s):  
Intersecting Family ◽  
Intersecting Families

Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is a fractional $L$-intersecting family if for every distinct $i,j \in [m]$, there exists an $\frac{a}{b} \in L$ such that $|A_i \cap A_j| \in \{ \frac{a}{b}|A_i|, \frac{a}{b} |A_j|\}$. In this paper, we introduce and study the notion of fractional $L$-intersecting families.

scholarly journals On the union of intersecting families

2019 ◽  
Vol 28 (06) ◽  
pp. 826-839
Author(s):  
David Ellis ◽  
Noam Lifshitz
Keyword(s):  
Fixed Integer ◽  
Empty Intersection ◽  
Intersecting Family ◽  
The Family ◽  
Intersecting Families ◽  
Element Set

AbstractA family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ⩾ 2 and for any $$k \le ({1 \over 2} - o(1))n$$, if X is an n-element set, and $${\cal F} = {\cal F}_1 \cup {\cal F}_2 \cup \cdots \cup {\cal F}_r $$, where each $$ {\cal F}_i $$ is an intersecting family of k-element subsets of X, then $$|{\cal F}| \le \left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right)$$, with equality only if $${\cal F} = \{ S \subset X:|S| = k,\;S \cap R \ne \emptyset \} $$ for some R ⊂ X with |R| = r. This is best possible up to the size of the o(1) term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis $$k &#x003C; (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

Intersecting Families are Essentially Contained in Juntas

2009 ◽  
Vol 18 (1-2) ◽  
pp. 107-122 ◽  
Author(s):  
IRIT DINUR ◽  
EHUD FRIEDGUT
Keyword(s):  
Fourier Analysis ◽  
Boolean Functions ◽  
Simple Description ◽  
Discrete Fourier Analysis ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Positive Integers

A family$\J$of subsets of {1, . . .,n} is called aj-junta if there existsJ⊆ {1, . . .,n}, with |J| =j, such that the membership of a setSin$\J$depends only onS∩J.In this paper we provide a simple description of intersecting families of sets. Letnandkbe positive integers withk<n/2, and let$\A$be a family of pairwise intersecting subsets of {1, . . .,n}, all of sizek. We show that such a family is essentially contained in aj-junta$\J$, wherejdoes not depend onnbut only on the ratiok/nand on the interpretation of ‘essentially’.Whenk=o(n) we prove that every intersecting family ofk-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersecting family): for any such intersecting family$\A$there exists an elementi∈ {1, . . .,n} such that the number of sets in$\A$that do not containiis of order$\C {n-2}{k-2}$(which is approximately$\frac {k}{n-k}$times the size of a maximal intersecting family).Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.

scholarly journals Intersecting Families in a Subset of Boolean Lattices

10.37236/724 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jun Wang ◽  
Huajun Zhang
Keyword(s):  
Intersecting Family ◽  
Intersecting Families ◽  
Boolean Lattices ◽  
Disjoint Sets ◽  
Positive Integers

Let $n, r$ and $\ell$ be distinct positive integers with $r < \ell\leq n/2$, and let $X_1$ and $X_2$ be two disjoint sets with the same size $n$. Define $$\mathcal{F}=\left\{A\in \binom{X}{r+\ell}: \mbox{$|A\cap X_1|=r$ or $\ell$}\right\},$$ where $X=X_1\cup X_2$. In this paper, we prove that if $\mathcal{S}$ is an intersecting family in $\mathcal{F}$, then $|\mathcal{S}|\leq \binom{n-1}{r-1}\binom{n}{\ell}+\binom{n-1}{\ell-1}\binom{n}{r}$, and equality holds if and only if $\mathcal{S}=\{A\in\mathcal{F}: a\in A\}$ for some $a\in X$.

scholarly journals A New Construction of Non-Extendable Intersecting Families of Sets

10.37236/5976 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Kaushik Majumder
Keyword(s):  
Short Note ◽  
Natural Logarithm ◽  
Intersecting Family ◽  
Intersecting Families ◽  
New Construction

In 1975, Lovász conjectured that any maximal intersecting family of $k$-sets has at most $\lfloor(e-1)k!\rfloor$ blocks, where $e$ is the base of the natural logarithm. This conjecture was disproved in 1996 by Frankl and his co-authors. In this short note, we reprove the result of Frankl et al. using a vastly simplified construction of maximal intersecting families with many blocks. This construction yields a maximal intersecting family $\mathbb{G}_{k}$ of $k-$sets whose number of blocks is asymptotic to $e^{2}(\frac{k}{2})^{k-1}$ as $k\rightarrow\infty$.

scholarly journals Stability Analysis for $k$-wise Intersecting Families

10.37236/602 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vikram Kamat
Keyword(s):  
Stability Analysis ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Circle Method ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Element Set

We consider the following generalization of the seminal Erdős–Ko–Rado theorem, due to Frankl. For some $k\geq 2$, let $\mathcal{F}$ be a $k$-wise intersecting family of $r$-subsets of an $n$ element set $X$, i.e. for any $F_1,\ldots,F_k\in \mathcal{F}$, $\cap_{i=1}^k F_i\neq \emptyset$. If $r\leq \dfrac{(k-1)n}{k}$, then $|\mathcal{F}|\leq {n-1 \choose r-1}$. We prove a stability version of this theorem, analogous to similar results of Dinur-Friedgut, Keevash-Mubayi and others for the EKR theorem. The technique we use is a generalization of Katona's circle method, initially employed by Keevash, which uses expansion properties of a particular Cayley graph of the symmetric group.

scholarly journals Compressions and Probably Intersecting Families

2012 ◽  
Vol 21 (1-2) ◽  
pp. 301-313 ◽  
Author(s):  
PAUL A. RUSSELL
Keyword(s):  
Main Tool ◽  
Independent Interest ◽  
Compression Method ◽  
Simple Fact ◽  
New Family ◽  
Fixed Probability ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Nested Sequence

A family of sets is said to be intersecting if A ∩ B ≠ ∅ for all A, B ∈ . It is a well-known and simple fact that an intersecting family of subsets of [n] = {1, 2, . . ., n} can contain at most 2n−1 sets. Katona, Katona and Katona ask the following question. Suppose instead ⊂ [n] satisfies || = 2n−1 + i for some fixed i > 0. Create a new family p by choosing each member of independently with some fixed probability p. How do we choose to maximize the probability that p is intersecting? They conjecture that there is a nested sequence of optimal families for i = 1, 2,. . ., 2n−1. In this paper, we show that the families [n](≥r) = {A ⊂ [n]: |A| ≥ r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there exist optimal families lying between those we have found. It turns out that the optimal families we find simultaneously maximize the number of intersecting subfamilies of each possible order.Standard compression techniques appear inadequate to solve the problem as they do not preserve intersection properties of subfamilies. Instead, our main tool is a novel compression method, together with a way of ‘compressing subfamilies’, which may be of independent interest.

On the Erdős–Ginzburg–Ziv invariant and zero-sum Ramsey number for intersecting families

2014 ◽  
Vol 10 (07) ◽  
pp. 1637-1647 ◽  
Author(s):  
Haiyan Zhang ◽  
Guoqing Wang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Zero Sum

Let G be a finite abelian group, and let m > 0 with exp (G) | m. Let sm(G) be the generalized Erdős–Ginzburg–Ziv invariant which denotes the smallest positive integer d such that any sequence of elements in G of length d contains a subsequence of length m with sum zero in G. For any integer r > 0, let [Formula: see text] be the collection of all r-uniform intersecting families of size m. Let [Formula: see text] be the smallest positive integer d such that any G-coloring of the edges of the complete r-uniform hypergraph [Formula: see text] yields a zero-sum copy of some intersecting family in [Formula: see text]. Among other results, we mainly prove that [Formula: see text], where Ω(sm(G)) denotes the least positive integer n such that [Formula: see text], and we show that if r | Ω(sm(G)) – 1 then [Formula: see text].

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