scholarly journals An upper bound for the size of a k-uniform intersecting family with covering number k

2017 ◽  
Vol 147 ◽  
pp. 18-26 ◽  
Author(s):  
Andrii Arman ◽  
Troy Retter
Keyword(s):  
Upper Bound ◽  
Covering Number ◽  
Intersecting Family

scholarly journals Set covering number for a finite set

1996 ◽  
Vol 53 (2) ◽  
pp. 267-269
Author(s):  
H.-C. Chang ◽  
N. Prabhu
Keyword(s):  
Upper Bound ◽  
Covering Number ◽  
Set Covering ◽  
Finite Set ◽  
Minimum Number

Given a finite set S of cardinality N, the minimum number of j-subsets of S needed to cover all the r-subsets of S is called the covering number C(N, j, r). While Erdös and Hanani's conjecture that was proved by Rödl, no nontrivial upper bound for C(N, j, r) was known for finite N. In this note we obtain a nontrivial upper bound by showing that for finite N,

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 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 indices of maximal subgroups and the normal primary coverings of finite groups

2019 ◽  
Vol 22 (6) ◽  
pp. 1015-1034
Author(s):  
Francesco Fumagalli
Keyword(s):  
Positive Integer ◽  
Finite Groups ◽  
Upper Bound ◽  
Prime Numbers ◽  
Covering Number ◽  
Arithmetic Functions ◽  
Maximal Subgroups ◽  
Theoretical Terms ◽  
Open Questions ◽  
Pure Number

Abstract We define and study two arithmetic functions {\gamma_{0}} and η, having domain the set of all finite groups whose orders are not prime powers. Namely, if G is such a group, we call {\gamma_{0}(G)} the normal primary covering number of G; this is defined as the smallest positive integer k such that the set of primary elements of G is covered by k conjugacy classes of proper (pairwise non-conjugate) subgroups of G. Also we set {\eta(G)} , the indices covering number of G, to be the smallest positive integer h such that G has h proper subgroups having coprime indices. This second function is an upper bound for {\gamma_{0}} , and it is much friendlier. The study of these functions for arbitrary finite groups reduces immediately to the non-abelian simple ones. We therefore apply CFSG to obtain bounds and interesting properties for {\gamma_{0}} and η. Open questions on these functions are reformulated in pure number-theoretical terms and lead to problems concerning the distributions and the representations of prime numbers.

Sign in / Sign up

Export Citation Format

Share Document