On sperner families in which nok sets have an empty intersection III

COMBINATORICA ◽  
1982 ◽  
Vol 2 (1) ◽  
pp. 25-36 ◽  
Author(s):  
H. -D. O. F. Gronau
Keyword(s):  
Empty Intersection

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.

On Erdős–Ko–Rado for random hypergraphs I

2019 ◽  
Vol 28 (06) ◽  
pp. 881-916 ◽  
Author(s):  
A. Hamm ◽  
J. Kahn
Keyword(s):  
Empty Intersection ◽  
Image Position ◽  
Random Hypergraphs

AbstractA family of sets is intersecting if no two of its members are disjoint, and has the Erdős–Ko–Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty intersection.Denote by ${{\cal H}_k}(n,p)$ the random family in which each k-subset of {1, …, n} is present with probability p, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks: \begin{equation} {\rm{For what }}p = p(n,k){\rm{is}}{{\cal H}_k}(n,p){\rm{likely to be EKR}}? \end{equation} Here, for fixed c < 1/4, and $k \lt \sqrt {cn\log n} $ we give a precise answer to this question, characterizing those sequences p = p(n, k) for which \begin{equation} {\mathbb{P}}({{\cal H}_k}(n,p){\rm{is EKR}}{\kern 1pt} ) \to 1{\rm{as }}n \to \infty . \end{equation}

On the maximal numerical range of some matrices

2018 ◽  
Vol 34 ◽  
pp. 288-303 ◽  
Author(s):  
Ali Hamed ◽  
Ilya Spitkovsky
Keyword(s):  
Numerical Range ◽  
Maximal Eigenvalue ◽  
Direct Sums ◽  
Codimension One ◽  
Empty Intersection ◽  
Insight Into

The maximal numerical range $W_0(A)$ of a matrix $A$ is the (regular) numerical range $W(B)$ of its compression $B$ onto the eigenspace $\mathcal L$ of $A^*A$ corresponding to its maximal eigenvalue. So, always $W_0(A)\subseteq W(A)$. Conditions under which $W_0(A)$ has a non-empty intersection with the boundary of $W(A)$ are established, in particular, when $W_0(A)=W(A)$. The set $W_0(A)$ is also described explicitly for matrices unitarily similar to direct sums of $2$-by-$2$ blocks, and some insight into the behavior of $W_0(A)$ is provided when $\mathcal L$ has codimension one.

n-FOLD TIME COINCIDENCES ON IGEC GRAVITATIONAL WAVE EVENTS DATA USING DATA-BASE TECHNOLOGY

2000 ◽  
Vol 09 (03) ◽  
pp. 347-351
Author(s):  
ROBERTO TERENZI
Keyword(s):  
Gravitational Wave ◽  
Data Base ◽  
Data Retrieval ◽  
Analysis Data ◽  
Time Interval ◽  
Empty Intersection ◽  
Coincidence Set ◽  
Data Base Management System ◽  
Base Management ◽  
Using Data

Gravitational events on IGEC data are marked with their detection time td. Although td takes into account phase shifts due to hardware and software, it could be statistically retarded or anticipated by noise with respect to the arrival time ta of a gravitational wave. Then we can estimate a time interval Δ for each event so that we can consider the ta of the event equally likely in Δ. Consequently we can define a n-fold coincidence as a n-tuple of events whose associated time intervals have a non-empty intersection. Using this definition and taking into account some time interval properties we show that a n-fold coincidence set can be computed by an appropriate selection of events from the 2-fold coincidence sets. Furthermore, we stress that a data base management system (DBMS) is a very effective tool to store IGEC data (i.e. events) and to implement n-fold coincidence analysis based on the above considerations. Finally we propose to use a DBMS to build a database of both events and n-fold coincidence analysis data in order to have an efficient and powerful tool for data retrieval and further analysis.

scholarly journals MAXIMIZING THE AREA OF OVERLAP OF TWO UNIONS OF DISKS UNDER RIGID MOTION

2009 ◽  
Vol 19 (06) ◽  
pp. 533-556 ◽  
Author(s):  
SERGIO CABELLO ◽  
MARK DE BERG ◽  
PANOS GIANNOPOULOS ◽  
CHRISTIAN KNAUER ◽  
RENÉ VAN OOSTRUM ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
High Probability ◽  
Rigid Motion ◽  
Maximum Area ◽  
Empty Intersection ◽  
Rigid Motions ◽  
Unit Disks

Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m ≥ n. We consider the problem of finding a translation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1 - ∊)-approximation algorithms for translations and for rigid motions, which run in O((nm/∊2) log (m/∊)) and O((n2m2/∊3) log m)) time, respectively. For rigid motions, we can also compute a (1 - ∊)-approximation in O((m2n4/3Δ1/3/∊3) log n log m) time, where Δ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1 - ∊)-approximation algorithm for rigid motions that runs in O((m2/∊4) log 2(m/∊) log m) time and succeeds with high probability. Our results generalize to the case where A and B consist of possibly intersecting disks of different radii, provided that (i) the ratio of the radii of any two disks in A ∪ B is bounded, and (ii) within each set, the maximum number of disks with a non-empty intersection is bounded.

Smallest Sets of Longest Paths with Empty Intersection

1996 ◽  
Vol 5 (4) ◽  
pp. 429-436 ◽  
Author(s):  
Z. Skupień
Keyword(s):  
Connected Graph ◽  
Connected Graphs ◽  
Minimal Sets ◽  
Empty Intersection ◽  
Longest Cycles ◽  
Longest Paths

It is shown that, for every integer v < 7, there is a connected graph in which some v longest paths have empty intersection, but any v – 1 longest paths have a vertex in common. Moreover, connected graphs having seven or five minimal sets of longest paths (longest cycles) with empty intersection are presented. A 26-vertex 2-connected graph whose longest paths have empty intersection is exhibited.

Integrating Self and System: An Empty Intersection?

1995 ◽  
Vol 34 (1) ◽  
pp. 21-44 ◽  
Author(s):  
Robert Rosenbaum ◽  
John Dyckman
Keyword(s):  
Empty Intersection

scholarly journals Families of partial representing sets

1985 ◽  
Vol 38 (2) ◽  
pp. 198-206
Author(s):  
Kevin P. Balanda
Keyword(s):  
Disjoint Family ◽  
Empty Intersection ◽  
The Family ◽  
Almost Disjoint ◽  
Almost Disjoint Family

AbstractAssume GCH. Let κ, μ, Σ be cardinals, with κ infinite. Let be a family consisting of λ pairwise almost disjoint subsets of Σ each of size κ, whose union is Σ. In this note it is shown that for each μ with 1 ≤ μ ≤min(λ, Σ), there is a “large” almost disjoint family of μ-sized subsets of Σ, each member of having non-empty intersection with at least μ members of the family .

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