scholarly journals Compressions and Probably Intersecting Families

2012 ◽  
Vol 21 (1-2) ◽  
pp. 301-313 ◽  
Author(s):  
PAUL A. RUSSELL
Keyword(s):  
Main Tool ◽  
Independent Interest ◽  
Compression Method ◽  
Simple Fact ◽  
New Family ◽  
Fixed Probability ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Nested Sequence

A family of sets is said to be intersecting if A ∩ B ≠ ∅ for all A, B ∈ . It is a well-known and simple fact that an intersecting family of subsets of [n] = {1, 2, . . ., n} can contain at most 2n−1 sets. Katona, Katona and Katona ask the following question. Suppose instead ⊂ [n] satisfies || = 2n−1 + i for some fixed i > 0. Create a new family p by choosing each member of independently with some fixed probability p. How do we choose to maximize the probability that p is intersecting? They conjecture that there is a nested sequence of optimal families for i = 1, 2,. . ., 2n−1. In this paper, we show that the families [n](≥r) = {A ⊂ [n]: |A| ≥ r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there exist optimal families lying between those we have found. It turns out that the optimal families we find simultaneously maximize the number of intersecting subfamilies of each possible order.Standard compression techniques appear inadequate to solve the problem as they do not preserve intersection properties of subfamilies. Instead, our main tool is a novel compression method, together with a way of ‘compressing subfamilies’, which may be of independent interest.

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 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.

scholarly journals Counting Intersecting and Pairs of Cross-Intersecting Families

2017 ◽  
Vol 27 (1) ◽  
pp. 60-68 ◽  
Author(s):  
PETER FRANKL ◽  
ANDREY KUPAVSKII
Keyword(s):  
Classical Result ◽  
Maximum Size ◽  
Extremal Combinatorics ◽  
Intersecting Family ◽  
Intersecting Families

A family of subsets of {1,. . .,n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko and Rado determines the maximum size of an intersecting family of k-subsets of {1,. . .,n}. In this paper we study the following problem: How many intersecting families of k-subsets of {1,. . .,n} are there? Improving a result of Balogh, Das, Delcourt, Liu and Sharifzadeh, we determine this quantity asymptotically for n ≥ 2k+2+2$\sqrt{k\log k}$ and k → ∞. Moreover, under the same assumptions we also determine asymptotically the number of non-trivial intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.

scholarly journals Fractional L-intersecting Families

10.37236/7846 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Niranjan Balachandran ◽  
Rogers Mathew ◽  
Tapas Kumar Mishra
Keyword(s):  
Intersecting Family ◽  
Intersecting Families

Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is a fractional $L$-intersecting family if for every distinct $i,j \in [m]$, there exists an $\frac{a}{b} \in L$ such that $|A_i \cap A_j| \in \{ \frac{a}{b}|A_i|, \frac{a}{b} |A_j|\}$. In this paper, we introduce and study the notion of fractional $L$-intersecting families.

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 $$k &#x003C; (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

scholarly journals Covering and tiling hypergraphs with tight cycles

2020 ◽  
pp. 1-42
Author(s):  
Jie Han ◽  
Allan Lo ◽  
Nicolás Sanhueza-Matamala
Keyword(s):  
Main Tool ◽  
Independent Interest ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Disjoint

Abstract A k-uniform tight cycle $C_s^k$ is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd (k, s) = 1 or k / gcd (k,s) is even. We prove that if $s \ge 2{k^2}$ and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V(H)|, then every vertex is covered by a copy of $C_s^k$ . The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order in which a tight path wraps around a complete k-partite k-uniform hypergraph, which may be of independent interest. For hypergraphs F and H, a perfect F-tiling in H is a spanning collection of vertex-disjoint copies of F. For $k \ge 3$ , there are currently only a handful of known F-tiling results when F is k-uniform but not k-partite. If s ≢ 0 mod k, then $C_s^k$ is not k-partite. Here we prove an F-tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for $s \ge 5{k^2}$ , every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect $C_s^k$ -tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

Intersecting Families are Essentially Contained in Juntas

2009 ◽  
Vol 18 (1-2) ◽  
pp. 107-122 ◽  
Author(s):  
IRIT DINUR ◽  
EHUD FRIEDGUT
Keyword(s):  
Fourier Analysis ◽  
Boolean Functions ◽  
Simple Description ◽  
Discrete Fourier Analysis ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Positive Integers

A family$\J$of subsets of {1, . . .,n} is called aj-junta if there existsJ⊆ {1, . . .,n}, with |J| =j, such that the membership of a setSin$\J$depends only onS∩J.In this paper we provide a simple description of intersecting families of sets. Letnandkbe positive integers withk<n/2, and let$\A$be a family of pairwise intersecting subsets of {1, . . .,n}, all of sizek. We show that such a family is essentially contained in aj-junta$\J$, wherejdoes not depend onnbut only on the ratiok/nand on the interpretation of ‘essentially’.Whenk=o(n) we prove that every intersecting family ofk-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersecting family): for any such intersecting family$\A$there exists an elementi∈ {1, . . .,n} such that the number of sets in$\A$that do not containiis of order$\C {n-2}{k-2}$(which is approximately$\frac {k}{n-k}$times the size of a maximal intersecting family).Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.

scholarly journals Cyclic Flats of a Polymatroid

2020 ◽  
Vol 24 (4) ◽  
pp. 637-648
Author(s):  
Laszlo Csirmaz
Keyword(s):  
Main Tool ◽  
Rank Function ◽  
Independent Interest ◽  
Discrete Measure

Abstract Polymatroids can be considered as “fractional matroids” where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a polymatroid carefully, the characterization by Bonin and de Mier of the ranked lattice of cyclic flats carries over to polymatroids. The main tool, which might be of independent interest, is a convolution-like method which creates a polymatroid from a ranked lattice and a discrete measure. Examples show the ease of using the convolution technique.

On radical extensions of rings

1967 ◽  
Vol 7 (4) ◽  
pp. 552-554 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Jacobson Radical ◽  
Main Tool ◽  
Independent Interest ◽  
Minimum Condition ◽  
Radical Extension ◽  
Commutativity Theorems

A ring K is a radical extension of a subring B if for each x ∈ K there is aninteger n = n(x) > 0 such that xn ∈ B. In [2] and [3], C. Faith considered radical extensions in connection with commutativity questions, as well as the generation of rings. In this paper additional commutativity theorems are established, and rings with right minimum condition are examined. The main tool is Theorem 1.1 which relates the Jacobson radical of K to that of B, and is of independent interest in itself. The author is indebted to the referee for his helpful suggestions, in particular for the strengthening of Theorem 2.1.

scholarly journals Intersecting Families in a Subset of Boolean Lattices

10.37236/724 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jun Wang ◽  
Huajun Zhang
Keyword(s):  
Intersecting Family ◽  
Intersecting Families ◽  
Boolean Lattices ◽  
Disjoint Sets ◽  
Positive Integers

Let $n, r$ and $\ell$ be distinct positive integers with $r < \ell\leq n/2$, and let $X_1$ and $X_2$ be two disjoint sets with the same size $n$. Define $$\mathcal{F}=\left\{A\in \binom{X}{r+\ell}: \mbox{$|A\cap X_1|=r$ or $\ell$}\right\},$$ where $X=X_1\cup X_2$. In this paper, we prove that if $\mathcal{S}$ is an intersecting family in $\mathcal{F}$, then $|\mathcal{S}|\leq \binom{n-1}{r-1}\binom{n}{\ell}+\binom{n-1}{\ell-1}\binom{n}{r}$, and equality holds if and only if $\mathcal{S}=\{A\in\mathcal{F}: a\in A\}$ for some $a\in X$.

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