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 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.

scholarly journals Stability Analysis for $k$-wise Intersecting Families

10.37236/602 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vikram Kamat
Keyword(s):  
Stability Analysis ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Circle Method ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Element Set

We consider the following generalization of the seminal Erdős–Ko–Rado theorem, due to Frankl. For some $k\geq 2$, let $\mathcal{F}$ be a $k$-wise intersecting family of $r$-subsets of an $n$ element set $X$, i.e. for any $F_1,\ldots,F_k\in \mathcal{F}$, $\cap_{i=1}^k F_i\neq \emptyset$. If $r\leq \dfrac{(k-1)n}{k}$, then $|\mathcal{F}|\leq {n-1 \choose r-1}$. We prove a stability version of this theorem, analogous to similar results of Dinur-Friedgut, Keevash-Mubayi and others for the EKR theorem. The technique we use is a generalization of Katona's circle method, initially employed by Keevash, which uses expansion properties of a particular Cayley graph of the symmetric group.

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 Most Probably Intersecting Families of Subsets

2012 ◽  
Vol 21 (1-2) ◽  
pp. 219-227 ◽  
Author(s):  
GYULA O. H. KATONA ◽  
GYULA Y. KATONA ◽  
ZSOLT KATONA
Keyword(s):  
Exact Solution ◽  
Maximum Size ◽  
Proper Subset ◽  
Analogous Problem ◽  
New Family ◽  
Intersecting Family ◽  
The Family ◽  
Intersecting Families ◽  
The One ◽  
Element Set

Let be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality || ≤ 2n−1. Suppose that ||=2n−1 + i. Choose the members of independently with probability p (delete them with probability 1 − p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all $\lfloor \frac{n}{ 2}\rfloor$-element subsets. We determine the most probably inclusion-free family too, when the number of members is $\binom{n}{ \lfloor \frac{n}{ 2}\rfloor} +1$.

scholarly journals Two-Part Set Systems

10.37236/2067 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Péter L. Erdős ◽  
Dániel Gerbner ◽  
Nathan Lemons ◽  
Dhruv Mubayi ◽  
Cory Palmer ◽  
...  
Keyword(s):  
Intersection Property ◽  
Independent Interest ◽  
Sperner Property ◽  
Set Systems ◽  
Intersecting Families ◽  
Element Set ◽  
The Way ◽  
Families Of Subsets

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\mathcal{F}$ is a family of subsets of $X$ such that  no two sets $A, B \in \mathcal{F}$  satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B\cap X_i$ for some $i$, then $|\mathcal{F}| \le {n \choose \lfloor n/2\rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following  new result which may be of independent interest: let $\mathcal{F},\mathcal{G}$ be intersecting families of subsets of an $n$-element set that are additionally cross-Sperner, meaning that if $A \in\mathcal{F}$ and $B \in \mathcal{G}$, then $A \not\subset B$ and $B \not\subset A$. Then  $|\mathcal{F}| +|\mathcal{G}| \le 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.

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 Probably Intersecting Families are Not Nested

2012 ◽  
Vol 22 (1) ◽  
pp. 146-160 ◽  
Author(s):  
PAUL A. RUSSELL ◽  
MARK WALTERS
Keyword(s):  
Intersecting Family ◽  
Intersecting Families ◽  
Nested Sequence ◽  
Element Set

It is well known that an intersecting family of subsets of an n-element set can contain at most 2n−1 sets. It is natural to wonder how ‘close’ to intersecting a family of size greater than 2n−1 can be. Katona, Katona and Katona introduced the idea of a ‘most probably intersecting family’. Suppose that is a family and that 0 < p < 1. Let (p) be the (random) family formed by selecting each set in independently with probability p. A family is most probably intersecting if it maximizes the probability that (p) is intersecting over all families of size ||.Katona, Katona and Katona conjectured that there is a nested sequence consisting of most probably intersecting families of every possible size. We show that this conjecture is false for every value of p provided that n is sufficiently large.

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.

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