scholarly journals Almost Intersecting Families

10.37236/9609 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Peter Frankl ◽  
Andrey Kupavskii
Keyword(s):  
The Family ◽  
Intersecting Families

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of $[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \varnothing$ for all $F, F' \in \mathcal F$. It is called almost intersecting if it is not intersecting but to every $F \in \mathcal F$ there is at most one $F'\in \mathcal F$ satisfying $F \cap F' = \varnothing$. Gerbner et al. proved that if $n \geq 2k + 2$ then $|\mathcal F| \leqslant {n - 1\choose k - 1}$ holds for almost  intersecting families. Our main result implies the considerably stronger and best possible bound $|\mathcal F| \leqslant {n - 1\choose k - 1} - {n - k - 1\choose k - 1} + 2$ for $n > (2 + o(1))k$, $k\ge 3$.

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.

scholarly journals Cross-Intersecting Families of Labeled Sets

10.37236/3047 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Huajun Zhang
Keyword(s):  
The Family ◽  
Intersecting Families ◽  
Positive Integers

For two positive integers $n$ and $p$, let $\mathcal{L}_{p}$ be the family of labeled $n$-sets given by $$\mathcal{L}_{p}=\big\{\{(1,\ell_1),(2,\ell_2),\ldots,(n,\ell_n)\}: \ell_i\in[p], i=1,2\ldots,n\big\}.$$ Families $\mathcal{A}$ and $\mathcal{B}$ are said to be cross-intersecting if $A\cap B\neq\emptyset$ for all $A\in \mathcal{A}$ and $B\in\mathcal{B}$. In this paper, we will prove that for $p\geq 4$,  if $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting subfamilies of $\mathcal{L}_{\mathfrak{p}}$, then $|\mathcal{A}||\mathcal{B}|\leq p^{2n-2}$, and equality holds if and only if $\mathcal{A}$ and $\mathcal{B}$ are an identical largest intersecting subfamily of $\mathcal{L}_{p}$.

scholarly journals A Sharp Bound for the Product of Weights of Cross-Intersecting Families

10.37236/5782 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Peter Borg
Keyword(s):  
General Setting ◽  
Sharp Bound ◽  
Integer Sequences ◽  
Sharp Bounds ◽  
The Family ◽  
Intersecting Families

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$. For any two integers $n$ and $k$ with $1 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of subsets of $\{1, \dots, n\}$ of size at most $k$, and let $\mathcal{S}_{n,k}$ denote the family of sets in ${[n] \choose \leq k}$ that contain $1$. The author recently showed that if $\mathcal{A} \subseteq {[m] \choose \leq r}$, $\mathcal{B} \subseteq {[n] \choose \leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then $|\mathcal{A}||\mathcal{B}| \leq \mathcal{S}_{m,r}||\mathcal{S}_{n,s}|$. We prove a version of this result for the more general setting of \emph{weighted} sets. We show that if $g : {[m] \choose \leq r} \rightarrow \mathbb{R}^+$ and $h : {[n] \choose \leq s} \rightarrow \mathbb{R}^+$ are functions that obey certain conditions, $\mathcal{A} \subseteq {[m] \choose \leq r}$, $\mathcal{B} \subseteq {[n] \choose \leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then $$\sum_{A \in \mathcal{A}} g(A) \sum_{B \in \mathcal{B}} h(B) \leq \sum_{C \in \mathcal{S}_{m,r}} g(C) \sum_{D \in \mathcal{S}_{n,s}} h(D).$$The bound is attained by taking $\mathcal{A} = \mathcal{S}_{m,r}$ and $\mathcal{B} = \mathcal{S}_{n,s}$. We also show that this result yields new sharp bounds for the product of sizes of cross-intersecting families of integer sequences and of cross-intersecting families of multisets.

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 Intersecting and Cross-Intersecting Families of Labeled Sets

10.37236/884 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Peter Borg
Keyword(s):  
Sharp Bound ◽  
General Inequality ◽  
The Family ◽  
Intersecting Families ◽  
Special Case

A family ${\cal A}$ of sets is said to be intersecting if any two sets in ${\cal A}$ intersect. Families ${\cal A}_1, ..., {\cal A}_p$ are said to be cross-intersecting if, for any $i, j \in \{1, ..., p\}$ such that $i \neq j$, any set in ${\cal A}_i$ intersects any set in ${\cal A}_j$. For ${\bf k} = (k_1, ..., k_n) \in {\Bbb N}^n$, $2 \leq k_1 \leq ... \leq k_n$, let ${\cal L}_{\bf{k}}$ be the family of labeled $n$-sets given by ${\cal L}_{\bf{k}} := \{\{(1,l_1), ..., (n,l_n)\} \colon l_i \in \{1, ..., k_i\}, i = 1, ..., n\}$. We point out a relationship between intersecting families and cross-intersecting families of labeled sets, and we show that, if ${\cal A}_1, ..., {\cal A}_p$ are cross-intersecting sub-families of ${\cal L}_{\bf{k}}$, then $$ \sum_{j = 1}^p |{\cal A}_j| \leq \left\{ \matrix{ k_1k_2...k_n & \hbox{if $p \leq k_1$};\cr pk_2...k_n & \hbox{if $p \geq k_1$}.\cr } \right. $$ We also determine the cases of equality. We then obtain a more general inequality, a special case of which is a sharp bound for cross-intersecting families of permutations.

scholarly journals The Intersection Structure of $t$-Intersecting Families

10.37236/1985 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
John Talbot
Keyword(s):  
Upper Bound ◽  
Common Elements ◽  
Intersecting Family ◽  
The Family ◽  
Intersecting Families ◽  
Asymptotic Upper Bound

A family of sets is $t$-intersecting if any two sets from the family contain at least $t$ common elements. Given a $t$-intersecting family of $r$-sets from an $n$-set, how many distinct sets of size $k$ can occur as pairwise intersections of its members? We prove an asymptotic upper bound on this number that can always be achieved. This result can be seen as a generalization of the Erdős-Ko-Rado theorem.

Triassic sponges (Sphinctozoa) from Hells Canyon, Oregon

1988 ◽  
Vol 62 (03) ◽  
pp. 419-423 ◽  
Author(s):  
Baba Senowbari-Daryan ◽  
George D. Stanley
Keyword(s):  
Endemic Species ◽  
Upper Triassic ◽  
Island Arc ◽  
Tropical Island ◽  
The Family ◽  
Wallowa Terrane ◽  
Unique Condition ◽  
Hells Canyon

Two Upper Triassic sphinctozoan sponges of the family Sebargasiidae were recovered from silicified residues collected in Hells Canyon, Oregon. These sponges areAmblysiphonellacf.A. steinmanni(Haas), known from the Tethys region, andColospongia whalenin. sp., an endemic species. The latter sponge was placed in the superfamily Porata by Seilacher (1962). The presence of well-preserved cribrate plates in this sponge, in addition to pores of the chamber walls, is a unique condition never before reported in any porate sphinctozoans. Aporate counterparts known primarily from the Triassic Alps have similar cribrate plates but lack the pores in the chamber walls. The sponges from Hells Canyon are associated with abundant bivalves and corals of marked Tethyan affinities and come from a displaced terrane known as the Wallowa Terrane. It was a tropical island arc, suspected to have paleogeographic relationships with Wrangellia; however, these sponges have not yet been found in any other Cordilleran terrane.

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