scholarly journals The Intersection Structure of $t$-Intersecting Families

10.37236/1985 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
John Talbot
Keyword(s):  
Upper Bound ◽  
Common Elements ◽  
Intersecting Family ◽  
The Family ◽  
Intersecting Families ◽  
Asymptotic Upper Bound

A family of sets is $t$-intersecting if any two sets from the family contain at least $t$ common elements. Given a $t$-intersecting family of $r$-sets from an $n$-set, how many distinct sets of size $k$ can occur as pairwise intersections of its members? We prove an asymptotic upper bound on this number that can always be achieved. This result can be seen as a generalization of the Erdős-Ko-Rado theorem.

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 $$k < (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

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 Most Probably Intersecting Families of Subsets

2012 ◽  
Vol 21 (1-2) ◽  
pp. 219-227 ◽  
Author(s):  
GYULA O. H. KATONA ◽  
GYULA Y. KATONA ◽  
ZSOLT KATONA
Keyword(s):  
Exact Solution ◽  
Maximum Size ◽  
Proper Subset ◽  
Analogous Problem ◽  
New Family ◽  
Intersecting Family ◽  
The Family ◽  
Intersecting Families ◽  
The One ◽  
Element Set

Let be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality || ≤ 2n−1. Suppose that ||=2n−1 + i. Choose the members of independently with probability p (delete them with probability 1 − p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all $\lfloor \frac{n}{ 2}\rfloor$-element subsets. We determine the most probably inclusion-free family too, when the number of members is $\binom{n}{ \lfloor \frac{n}{ 2}\rfloor} +1$.

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.

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed

scholarly journals Communication Complexity And Linearly Ordered Sets

2015 ◽  
Vol 29 (1) ◽  
pp. 93-117
Author(s):  
Mieczysław Kula ◽  
Małgorzata Serwecińska
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Numerical Experiments ◽  
Communication Complexity ◽  
Ordered Sets ◽  
Finite Sets ◽  
The Family ◽  
Linear Lattices

AbstractThe paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to O(log2n), where n is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.

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.

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.

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