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Some results on intersecting families of subsets

Chuanzhong Zhu
1998 Discrete Mathematics  
The structure 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.  ... 
doi:10.1016/s0012-365x(96)00083-0 fatcat:j4rd7hxmm5duxk65lmpkxqzxfa

Extremal problems among subsets of a set

Paul Erdos, Daniel J. Kleitman
1974 Discrete Mathematics  
This paper is a survey 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 .  ... 
doi:10.1016/0012-365x(74)90140-x fatcat:ao42lqh2l5ff5ozkawxywtmimu

Extremal problems among subsets of a set

Paul Erdös, Daniel J. Kleitman
2006 Discrete Mathematics  
This paper is a survey 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 .  ... 
doi:10.1016/j.disc.2006.03.013 fatcat:rz7iin4j4fgyfmnb7epxw6ligu

Extremal set systems with restricted k-wise intersections

Zoltán Füredi, Benny Sudakov
2004 Journal of combinatorial theory. Series A  
A large variety 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  ... 
doi:10.1016/j.jcta.2003.10.008 fatcat:d4kfpkuvu5hc5okgdussyo5pv4

Maximal intersecting families

Aaron Meyerowitz
1995 European journal of combinatorics (Print)  
We give 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.  ... 
doi:10.1016/0195-6698(95)90004-7 fatcat:kprg6gl4wzf5dfsrwaoi4rysqu

Intersection theorems for multisets [article]

Karen Meagher, Alison Purdy
2015 arXiv   pre-print
We use graph homomorphisms and existing theorems for 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.  ... 
arXiv:1504.06657v2 fatcat:ikbzeewhpnhbhd5wyyeaomkaa4

A note on saturation for k-wise intersecting families [article]

Barnabás Janzer
2021 arXiv   pre-print
A familyof 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  ... 
arXiv:2111.12021v2 fatcat:osxhlggthndkndrfqqcp6e4vjm

A generalization of the Erdős-Ko-Rado Theorem [article]

Gábor Hegedüs
2015 arXiv   pre-print
Our main 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] .  ... 
arXiv:1512.05531v2 fatcat:fb7yrzvsbfb7zennw5lq3iusiq

Supports of continuous functions

Mark Mandelker
1971 Transactions of the American Mathematical Society  
A 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.  ... 
doi:10.1090/s0002-9947-1971-0275367-4 fatcat:oqrpr4ck35emna6ge4sqsw7kiu

Supports of Continuous Functions

Mark Mandelker
1971 Transactions of the American Mathematical Society  
A 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  ... 
doi:10.2307/1995598 fatcat:whgixjpn5vhlhd56yuqii3ma7i

Some intersection theorems for ordered sets and graphs

F.R.K Chung, R.L Graham, P Frankl, J.B Shearer
1986 Journal of combinatorial theory. Series A  
We will now use the Product Theorem to prove two theorems 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.  ... 
doi:10.1016/0097-3165(86)90019-1 fatcat:jyv3oagdk5fd5azhemeu5janru

Partitioning all k-subsets into r-wise intersecting families [article]

Noga Alon
2021 arXiv   pre-print
In the original arXiv version 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.  ... 
arXiv:2107.12741v2 fatcat:hm22siauzfbcbooyyjemf77yry

Intersection Theorems for t-Valued Functions

R.H. Schelp, M. Simonovits, V.T. Sós
1988 European journal of combinatorics (Print)  
More precisely, given a 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  ... 
doi:10.1016/s0195-6698(88)80049-0 fatcat:qqgz3drwh5gmzieqw7znyj2nhu

On well-filtered spaces and ordered sets

Xiaoyong Xi, Jimmie Lawson
2017 Topology and its Applications  
A topological space is well-filtered if any filtered 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.  ... 
doi:10.1016/j.topol.2017.06.002 fatcat:c4i4xmbobreyboosrtcrxddfyy

Some Erdős–Ko–Rado theorems for injections

Fiona Brunk, Sophie Huczynska
2010 European journal of combinatorics (Print)  
This paper investigates t-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.  ... 
doi:10.1016/j.ejc.2009.07.013 fatcat:yujvah3nvnhzhcg5ltqibiy3jy
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