scholarly journals Structural Results for Conditionally Intersecting Families and some Applications

10.37236/8894 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Xizhi Liu
Keyword(s):  
Upper Bounds ◽  
Empty Intersection ◽  
Intersecting Families

Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a question of Frankl, we present some structural results for families that are $(d,s)$-conditionally intersecting with $s\ge 2k+d-3$, and families that are $(k,2k)$-conditionally intersecting. As applications of our structural results we present some new proofs to the upper bounds for the size of the following $k$-uniform families on $[n]$: (a) $(d,2k+d-3)$-conditionally intersecting families with $n\ge 3k^5$; (b) $(k,2k)$-conditionally intersecting families with $n\ge k^2/(k-1)$; (c) Nonintersecting $(3,2k)$-conditionally intersecting families with $n\ge 3k\binom{2k}{k}$. Our results for $(c)$ confirms a conjecture of Mammoliti and Britz for the case $d=3$.

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.

Uniform Intersecting Families with Covering Number Restrictions

1998 ◽  
Vol 7 (1) ◽  
pp. 47-56
Author(s):  
P. FRANKL ◽  
K. OTA ◽  
N. TOKUSHIGE
Keyword(s):  
Upper Bounds ◽  
Covering Number ◽  
Intersecting Families

It is known that any k-uniform family with covering number t has at most ktt-covers. In this paper, we deal with intersecting families and give better upper bounds for the number of t-covers. Let pt(k) be the maximum number of t-covers in any k-uniform intersecting families with covering number t. We prove that, for a fixed t,formula hereIn the cases of t=4 and 5, we also prove that the coefficient of kt−1 in pt(k) is exactly (t2).

scholarly journals Erdős–Ko–Rado Theorem for a Restricted Universe

10.37236/8682 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Peter Frankl
Keyword(s):  
Upper Bounds ◽  
Intersecting Families ◽  
Element Set

A family $\mathcal F$ of $k$-element subsets of the $n$-element set $[n]$ is called \emph{intersecting} if $F \cap F'\neq \emptyset$ for all $F, F' \in \mathcal F$. In 1961 Erdős, Ko and Rado showed that $|\mathcal F| \leq {n - 1\choose k - 1}$ if $n \geq 2k$. Since then a large number of resultső providing best possible upper bounds on $|\mathcal F|$ under further restraints were proved. The paper of Li et al. is one of them. We consider the restricted universe $\mathcal W = \left\{F \in {[n]\choose k}: |F \cap [m]| \geq \ell \right\}$, $n \geq 2k$, $m \geq 2\ell$ and determine $\max |\mathcal F|$ for intersecting families $\mathcal F \subset \mathcal W$. Then we use this result to solve completely the problem considered by Li et al.

scholarly journals On intersecting families of finite sets

1980 ◽  
Vol 21 (3) ◽  
pp. 363-372 ◽  
Author(s):  
Peter Frankl
Keyword(s):  
Finite Sets ◽  
Empty Intersection ◽  
Intersecting Families

Let F be a family of k-element subsets of an n-set, n > n0(k). Suppose any two members of F have non-empty intersection. Let τ(F) denote min|T|, T meets every member of F. Erdös, Ko and Rado proved and that if equality holds then τ(F) = 1. Hilton and Milner determined max|F| for τ(F) = 2. In this paper we solve the problem for τ(F) = 3.The extremal families look quite complicated which shows the power of the methods used for their determination.

scholarly journals On the Union Complexity of Families of Axis-Parallel Rectangles with a Low Packing Number

10.37236/6792 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Chaya Keller ◽  
Shakhar Smorodinsky
Keyword(s):  
Upper Bounds ◽  
Packing Number ◽  
Empty Intersection ◽  
Axis Parallel

Let $\mathcal{R}$ be a family of $n$ axis-parallel rectangles with packing number $p-1$, meaning that among any $p$ of the rectangles, there are two with a non-empty intersection. We show that the union complexity of $\mathcal{R}$ is at most $O(n+p^2)$, and that the $(k-1)$-level complexity of $\mathcal{R}$ is at most $O(n+k p^2)$. Both upper bounds are tight.

Weighted multiply intersecting families

2003 ◽  
Vol 40 (3) ◽  
pp. 287-291 ◽  
Author(s):  
Peter Frankl ◽  
Norihide Tokushige
Keyword(s):  
Intersecting Families

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

Sign in / Sign up

Export Citation Format

Share Document