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Some results on intersecting families of subsets

1998
*
Discrete Mathematics
*

The structure

doi:10.1016/s0012-365x(96)00083-0
fatcat:j4rd7hxmm5duxk65lmpkxqzxfa
*of*this paper is as follow: In Sections 2 and 3, we prove*some**results*about*intersecting**families*for all n, which improves the*result**of*[ 1, 9] . ... Is}*intersecting**family*. Suppose that It is the minimum element*of*L such that IF,-N Fjl = It for*some*Fi, Fj E ~, t > 1. ...##
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Extremal problems among subsets of a set

1974
*
Discrete Mathematics
*

This paper is a survey

doi:10.1016/0012-365x(74)90140-x
fatcat:ao42lqh2l5ff5ozkawxywtmimu
*of*open problems and*results*involving extremal size*of*collections*of**subsets**of*a finite set subject to various restrictions, typically*on**intersections**of*members . ... Milner [33, 34] has*some**results**on*the first*of*these questions . The second is open . ... Among the maximal F's are*families*consisting*of*all*subsets*containing*some*single element . Such*families*have the property that all*intersections*are non-empty . ...##
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Extremal problems among subsets of a set

2006
*
Discrete Mathematics
*

This paper is a survey

doi:10.1016/j.disc.2006.03.013
fatcat:rz7iin4j4fgyfmnb7epxw6ligu
*of*open problems and*results*involving extremal size*of*collections*of**subsets**of*a finite set subject to various restrictions, typically*on**intersections**of*members . ... Let G4k be a*family*such that the*intersection**of*every k members is non-empty . We now describe*some**results*. No collection*of*non-disjoint*subsets*can contain a set and its complement . ... Milner [33, 34] has*some**results**on*the first*of*these questions . The second is open . ...##
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Extremal set systems with restricted k-wise intersections

2004
*
Journal of combinatorial theory. Series A
*

A large variety

doi:10.1016/j.jcta.2003.10.008
fatcat:d4kfpkuvu5hc5okgdussyo5pv4
*of*problems and*results*in Extremal Set Theory deal with estimates*on*the size*of*a*family**of*sets with*some*restrictions*on*the*intersections**of*its members. ... In this paper we obtain the following extension*of**some**of*these*results*when the restrictions apply to k-wise*intersections*, for k42: Let L be a*subset**of*non-negative integers*of*size s and let k42: ... Extending*some*earlier*results**of*Fisher, they proved that if F is a*family**of**subsets**of*an n-element set such that the*intersection**of*any two members*of*F has the same non-zero cardinality, then jFjpn ...##
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Maximal intersecting families

1995
*
European journal of combinatorics (Print)
*

We give

doi:10.1016/0195-6698(95)90004-7
fatcat:kprg6gl4wzf5dfsrwaoi4rysqu
*some*general*results**on*the collection*of*all such*families*for a fixed set and introduce a useful operation*of*switching which allows us to transform any*one*such*family*to another. ... We study maximum eardinality*families**of*pairwise*intersecting**subsets**of*an n-set. ... We let permutations in SYM(X) act*on**families**of**subsets*in the obvious way. We say that two maximal*intersecting**families*are isomorphic if*some*permutation maps*one*to the other. ...##
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Intersection theorems for multisets
[article]

2015
*
arXiv
*
pre-print

We use graph homomorphisms and existing theorems for

arXiv:1504.06657v2
fatcat:ikbzeewhpnhbhd5wyyeaomkaa4
*intersecting*and t-*intersecting*k-set systems to prove new*results*for*intersecting*and t-*intersecting**families**of*k-multisets. ... These*results*include a multiset version*of*the Hilton-Milner theorem and a theorem giving the size and structure*of*the largest t-*intersecting**family**of*k-multisets*of*an m-set when m ≤ 2k-t. ... In Section 1, we introduce notation and provide*some*background information*on*the Erdős-Ko-Rado theorem and the known*results*for*intersecting**families**of*multisets. ...##
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A note on saturation for k-wise intersecting families
[article]

2021
*
arXiv
*
pre-print

A

arXiv:2111.12021v2
fatcat:osxhlggthndkndrfqqcp6e4vjm
*family*ℱ*of**subsets**of*{1,... ... We show that for each k≥ 2 there is a maximal k-wise*intersecting**family**of*size O(2^n/(k-1)). ... (A*family*F*of**subsets**of*[n] is maximal k-wise*intersecting*if it is k-wise*intersecting*but no*family*F ′ over [n] strictly ...##
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A generalization of the Erdős-Ko-Rado Theorem
[article]

2015
*
arXiv
*
pre-print

Our main

arXiv:1512.05531v2
fatcat:fb7yrzvsbfb7zennw5lq3iusiq
*result*is a new upper bound for the size*of*k-uniform, L-*intersecting**families**of*sets, where L contains only positive integers. We characterize extremal*families*in this setting. ... Our proof is based*on*the Ray-Chaudhuri--Wilson Theorem. ... Introduction First we introduce*some*notations. Let [n] stand for the set {1, 2, . . . , n}. We denote the*family**of*all*subsets**of*[n] by 2 [n] . ...##
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Supports of continuous functions

1971
*
Transactions of the American Mathematical Society
*

A

doi:10.1090/s0002-9947-1971-0275367-4
fatcat:oqrpr4ck35emna6ge4sqsw7kiu
*family*&*of**subsets**of*a space X is said to be stable if every function in C(X) is bounded*on**some*member*of*#". ... We show that a completely regular Hausdorff space is realcompact if and only if every stable*family**of*closed*subsets*with the finite*intersection*property has nonempty*intersection*. ... Thus Jt is a*family**of*closed*subsets**of*X with the finite*intersection*property and empty*intersection*. Let fe CiX). Since/*(/?) is finite,/* is bounded*on**some*zero-set neighborhood W oí p in ßX. ...##
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Supports of Continuous Functions

1971
*
Transactions of the American Mathematical Society
*

A

doi:10.2307/1995598
fatcat:whgixjpn5vhlhd56yuqii3ma7i
*family*&*of**subsets**of*a space X is said to be stable if every function in C(X) is bounded*on**some*member*of*#". ... We show that a completely regular Hausdorff space is realcompact if and only if every stable*family**of*closed*subsets*with the finite*intersection*property has nonempty*intersection*. ... If X is dense in*some*Hausdorff realcompact space T, then every real continuous function*on*T is also constant ; hence T is pseudocompact, hence compact, and the subspace Zis completely regular, which ...##
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Some intersection theorems for ordered sets and graphs

1986
*
Journal of combinatorial theory. Series A
*

We will now use the Product Theorem to prove two theorems

doi:10.1016/0097-3165(86)90019-1
fatcat:jyv3oagdk5fd5azhemeu5janru
*on**intersection**families**of*graphs. THEOREM 8. ... In this paper, we consider the following general question: For a given*family*B*of**subsets**of*[n] = { 1, 2,..., n}, what is the largest*family*F*of**subsets**of*[n] satsifying F,F'EF-FnFzB for*some*BE B. ...##
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Partitioning all k-subsets into r-wise intersecting families
[article]

2021
*
arXiv
*
pre-print

In the original arXiv version

arXiv:2107.12741v2
fatcat:hm22siauzfbcbooyyjemf77yry
*of*this note we suggested a conjecture that the*family**of*all k-*subsets**of*an n-set cannot be partitioned into fewer than ⌈ n-r/r-1(k-1) ⌉ r-wise*intersecting**families*. ... We have recently learned, however, that the assertion*of*the conjecture for all values*of*the parameters follows from a recent*result**of*Azarpendar and Jafari . ... The main purpose*of*this brief note is to study the following extension*of*this*result*. Call a*family**of**subsets*r-wise*intersecting*if any collection*of*at most r*subsets*in it has a common point. ...##
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Intersection Theorems for t-Valued Functions

1988
*
European journal of combinatorics (Print)
*

More precisely, given a

doi:10.1016/s0195-6698(88)80049-0
fatcat:qqgz3drwh5gmzieqw7znyj2nhu
*family*:JI*of*k-eIement*subsets**of*S, it is assumed for each pair h, g E !F that there exists a B in :JI such that h = g*on*B. ... This paper investigates the maximum possible size*of**families*!F*of*I-valued functions*on*an n-element set S = {I, 2, . .. , n}, assuming any two functions*of*!F agree in sufficiently many places. ... If d is a*family**of**subsets**of*S such that the*intersection**of*each pair in d contains an element*of*each member*of**some*progression P, then Idl ~ 2 nk • Clearly, when d = 1 and t = 2 the*results**of*Theorem ...##
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On well-filtered spaces and ordered sets

2017
*
Topology and its Applications
*

A topological space is well-filtered if any filtered

doi:10.1016/j.topol.2017.06.002
fatcat:c4i4xmbobreyboosrtcrxddfyy
*family**of*compact sets with*intersection*in an open set must have*some*member*of*the*family*contained in the open set. ... Our main*results*focus*on*giving general sufficient conditions for a T 0 -space to be well-filtered, particularly the important case*of*directed complete partially ordered sets equipped with the Scott ... A nonempty*family**of**subsets**of*a set X is said to be filtered if any two members*of*the*family*contain*some*third member. ...##
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Some Erdős–Ko–Rado theorems for injections

2010
*
European journal of combinatorics (Print)
*

This paper investigates t-

doi:10.1016/j.ejc.2009.07.013
fatcat:yujvah3nvnhzhcg5ltqibiy3jy
*intersecting**families**of*injections, where We prove that if F is a 1-*intersecting*injection*family**of*maximal size then all elements*of*F have a fixed image point in common. ... We show that when n is large in terms*of*k and t, the set*of*injections which fix the first t points is the only t-*intersecting*injection*family**of*maximal size, up to permutations*of*[k] and [n]. ... If n ≥ 2 and F is an*intersecting**subset**of*S n with |F | = (n − 1)! then F is equivalent to the fix-*family*. This*result*has inspired numerous investigations*of**intersecting*permutation*families*. ...
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