Kleitman theorem for no 𝑠 pairwise disjoint sets

2018 ◽  
pp. 37-41
Keyword(s):  
Pairwise Disjoint ◽  
Disjoint Sets

scholarly journals Families with no s pairwise disjoint sets

2017 ◽  
Vol 95 (3) ◽  
pp. 875-894 ◽  
Author(s):  
Peter Frankl ◽  
Andrey Kupavskii
Keyword(s):  
Pairwise Disjoint ◽  
Disjoint Sets

Strongs-Sequences and Variations on Martin's Axiom

1985 ◽  
Vol 37 (4) ◽  
pp. 730-746 ◽  
Author(s):  
Juris Steprāns
Keyword(s):  
Pairwise Disjoint ◽  
Disjoint Family ◽  
Uncountable Family ◽  
Martin’S Axiom ◽  
Finite Set ◽  
Almost Disjoint ◽  
Disjoint Sets ◽  
Almost Disjoint Family ◽  
Require Equality ◽  
Single Set

As part of their study of βω — ω and βω1 — ω1, A. Szymanski and H. X. Zhou [3] were able to exploit the following difference between ω, and ω: ω1, contains uncountably many disjoint sets whereas any uncountable family of subsets of ω is, at best, almost disjoint. To translate this distinction between ω1, and ω to a possible distinction between βω1 — ω1, and βω — ω they used the fact that if a pairwise disjoint family of sets and a subset of each member of is chosen then it is trivial to find a single set whose intersection with each member is the chosen set. However, they noticed, it is not clear that the same is true if is only a pairwise almost disjoint family even if we only require equality except on a finite set. But any homeomorphism from βω1 — ω1 to βω — ω would have to carry a disjoint family of subsets of ω1, to an almost disjoint family of subsets of ω with this property. This observation should motivate the following definition.

scholarly journals Fitting classes and lattice formations I

2004 ◽  
Vol 76 (1) ◽  
pp. 93-108 ◽  
Author(s):  
M. Arroyo-Jordá ◽  
M. D. Pérez-Ramos
Keyword(s):  
Direct Product ◽  
Pairwise Disjoint ◽  
Nilpotent Groups ◽  
Saturated Formation ◽  
Closure Properties ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
Disjoint Sets ◽  
Lattice Formation ◽  
Fitting Classes

AbstractA lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving F-subnormal subgroups, for a lattice formation F of full characteristic, are studied. For a subgroup-closed saturated formation G, a characterisation of the G-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the N-projectors, N being the class of nilpotent groups.

Two Isomorphic 9 Order Steiner Triple Systems Large Sets

2013 ◽  
Vol 427-429 ◽  
pp. 1237-1240
Author(s):  
Zhao Di Xu ◽  
Xiao Yi Li ◽  
Wan Xi Chou
Keyword(s):  
Pairwise Disjoint ◽  
Permutation Matrix ◽  
Large Set ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Large Sets ◽  
Initial Block ◽  
Disjoint Sets ◽  
Basic Ideas ◽  
Block Permutation

This paper Clarifies the basic ideas of constructing the v order Steiner triple systems. This paper proposed the construction method of pairwise disjoint sets s(i)(v) for Steiner triple systems based on the initial block permutation matrix. And a method of initial block permutation matrix is given. This paper also introduced the entire construction process of two isomorphic 9 order Steiner triple systems large set. At last, this paper proved the number of pairwise disjoint forsi(9)is d(9)=7 .

scholarly journals On Weighted Pál Type (0,2)-interpolation on the Unit Circle

2021 ◽  
Author(s):  
Swarnima Bahadur ◽  
Sariya Bano
Keyword(s):  
Unit Circle ◽  
Legendre Polynomial ◽  
Pairwise Disjoint ◽  
Explicit Representation ◽  
Disjoint Sets ◽  
Ams Classification

Abstract In this paper, we study the explicit representation of weighted Pál-type (0,2)-interpolation on two pairwise disjoint sets of nodes on the unit circle, which are obtained by projecting vertically the zeros of (1−x2)Pn(x) and Pn′′(x) respectively, where Pn(x) stands for nth Legendre polynomial.AMS Classification (2000): 41A05, 30E10.

scholarly journals Chromatic Numbers of Stable Kneser Hypergraphs via Topological Tverberg-Type Theorems

2018 ◽  
Vol 2020 (13) ◽  
pp. 4037-4061 ◽  
Author(s):  
Florian Frick
Keyword(s):  
Euclidean Space ◽  
Chromatic Number ◽  
Lower Bounds ◽  
Pairwise Disjoint ◽  
Convex Hulls ◽  
Chromatic Numbers ◽  
Set Systems ◽  
Equivariant Topology ◽  
Disjoint Sets ◽  
Combination Of Methods

Abstract Kneser’s 1955 conjecture—proven by Lovász in 1978—asserts that in any partition of the $k$-subsets of $\{1, 2, \dots , n\}$ into $n-2k+1$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser’s conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász’ result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser’s conjecture. Here we prove a common generalization of all of these results. Moreover, we prove additional strengthenings by determining the chromatic number of certain sparse stable Kneser hypergraphs, and further develop a general approach to establishing lower bounds for chromatic numbers of hypergraphs using a combination of methods from equivariant topology and intersection results for convex hulls of points in Euclidean space.

On sets not belonging to algebras

2007 ◽  
Vol 72 (2) ◽  
pp. 483-500 ◽  
Author(s):  
L. Š. Grinblat
Keyword(s):  
Pairwise Disjoint ◽  
Finite Sequence ◽  
Countable Sequence ◽  
Finite Sequences ◽  
Disjoint Sets ◽  
Algebras Of Sets

AbstractLet be a finite sequence of algebras of sets given on a set with more than pairwise disjoint sets not belonging to It was shown in [4] and [5] that in this case Let us consider, instead a finite sequence of algebras It turns out that if for each natural i ≤ l there exist no less than pairwise disjoint sets not belonging to then But if l ≥ 195 and if for each natural i ≤ l there exist no less than pairwise disjoint sets not belonging to then After consideration of finite sequences of algebras, it is natural to consider countable sequences of algebras. We obtained two essentially important theorems on a countable sequence of almost σ-algebras (the concept of almost σ-algebra was introduced in [4]).

scholarly journals On the Number of Perfect Matchings in Random Lifts

2010 ◽  
Vol 19 (5-6) ◽  
pp. 791-817 ◽  
Author(s):  
CATHERINE GREENHILL ◽  
SVANTE JANSON ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Asymptotic Formula ◽  
Complete Graph ◽  
Pairwise Disjoint ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Lattice Points ◽  
Laplace’S Method ◽  
Second Moment ◽  
Multiple Dimensions ◽  
Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

Sign in / Sign up

Export Citation Format

Share Document