scholarly journals A New Upper Bound for Cancellative Pairs

10.37236/7210 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Barnabás Janzer
Keyword(s):  
Upper Bound ◽  
Set Systems ◽  
Element Set ◽  
Families Of Subsets

A pair $(\mathcal{A},\mathcal{B})$ of families of subsets of an $n$-element set is called cancellative if whenever $A,A'\in\mathcal{A}$ and $B\in\mathcal{B}$ satisfy $A\cup B=A'\cup B$, then $A=A'$, and whenever $A\in\mathcal{A}$ and $B,B'\in\mathcal{B}$ satisfy $A\cup B=A\cup B'$, then $B=B'$. It is known that there exist cancellative pairs with $|\mathcal{A}||\mathcal{B}|$ about $2.25^n$, whereas the best known upper bound on this quantity is $2.3264^n$. In this paper we improve this upper bound to $2.2682^n$. Our result also improves the best known upper bound for Simonyi's sandglass conjecture for set systems.

scholarly journals Set-Systems with Restricted Multiple Intersections

10.37236/1625 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Vince Grolmusz
Keyword(s):  
Upper Bound ◽  
Explicit Construction ◽  
Partial Answer ◽  
Residue Classes ◽  
Set Systems ◽  
Element Set ◽  
Multiple Intersections

We give a generalization for the Deza-Frankl-Singhi Theorem in case of multiple intersections. More exactly, we prove, that if ${\cal H}$ is a set-system, which satisfies that for some $k$, the $k$-wise intersections occupy only $\ell$ residue-classes modulo a $p$ prime, while the sizes of the members of ${\cal H}$ are not in these residue classes, then the size of ${\cal H}$ is at most $$(k-1)\sum_{i=0}^{\ell}{n\choose i}$$ This result considerably strengthens an upper bound of Füredi (1983), and gives partial answer to a question of T. Sós (1976). As an application, we give a direct, explicit construction for coloring the $k$-subsets of an $n$ element set with $t$ colors, such that no monochromatic complete hypergraph on $$\exp{(c(\log m)^{1/t}(\log \log m)^{1/(t-1)})}$$ vertices exists.

scholarly journals Set Systems with Restricted $t$-wise Intersections Modulo Prime Powers

10.37236/255 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rudy X. J. Liu
Keyword(s):  
Upper Bound ◽  
Prime Power ◽  
Set Systems ◽  
Composite Moduli ◽  
Element Set

We give a polynomial upper bound on the size of set systems with restricted $t$-wise intersections modulo prime powers. Let $t\geq 2$. Let $p$ be a prime and $q=p^{\alpha}$ be a prime power. Let ${\cal L}=\{l_1,l_2,\ldots,l_s\}$ be a subset of $\{0, 1, 2, \ldots, q-1\}$. If ${\cal F}$ is a family of subsets of an $n$ element set $X$ such that $|F_{1}\cap \cdots \cap F_{t}| \pmod{q} \in {\cal L}$ for any collection of $t$ distinct sets from ${\cal F}$ and $|F| \pmod{q} \notin {\cal L}$ for every $F\in {\cal F}$, then $$ |{\cal F}|\leq {t(t-1)\over2}\sum_{i=0}^{2^{s-1}}{n\choose i}. $$ Our result extends a theorem of Babai, Frankl, Kutin, and Štefankovič, who studied the $2$-wise case for prime power moduli, and also complements a result of Grolmusz that no polynomial upper bound holds for non-prime-power composite moduli.

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 An Erdős-Ko-Rado Type Theorem via the Polynomial Method

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 640
Author(s):  
Kyung-Won Hwang ◽  
Younjin Kim ◽  
Naeem N. Sheikh
Keyword(s):  
Upper Bound ◽  
Type Theorem ◽  
Nonempty Intersection ◽  
Polynomial Method ◽  
Linearly Independent ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Families Of Subsets

A family F is an intersecting family if any two members have a nonempty intersection. Erdős, Ko, and Rado showed that | F | ≤ n − 1 k − 1 holds for a k-uniform intersecting family F of subsets of [ n ] . The Erdős-Ko-Rado theorem for non-uniform intersecting families of subsets of [ n ] of size at most k can be easily proved by applying the above result to each uniform subfamily of a given family. It establishes that | F | ≤ n − 1 k − 1 + n − 1 k − 2 + ⋯ + n − 1 0 holds for non-uniform intersecting families of subsets of [ n ] of size at most k. In this paper, we prove that the same upper bound of the Erdős-Ko-Rado Theorem for k-uniform intersecting families of subsets of [ n ] holds also in the non-uniform family of subsets of [ n ] of size at least k and at most n − k with one more additional intersection condition. Our proof is based on the method of linearly independent polynomials.

scholarly journals Excluded subposets in the Boolean lattice

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gyula O.H. Katona
Keyword(s):  
Upper Bound ◽  
Boolean Lattice ◽  
The Other ◽  
Other Hand ◽  
International Audience ◽  
Element Set

International audience We are looking for the maximum number of subsets of an n-element set not containing 4 distinct subsets satisfying $A ⊂B, C ⊂B, C ⊂D$. It is proved that this number is at least the number of the $\lfloor \frac{n }{ 2}\rfloor$ -element sets times $1+\frac{2}{ n}$, on the other hand an upper bound is given with 4 replaced by the value 2.

scholarly journals Berge Cycles in Non-Uniform Hypergraphs

10.37236/9346 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Zoltán Füredi ◽  
Alexandr Kostochka ◽  
Ruth Luo
Keyword(s):  
Upper Bound ◽  
Minimum Degree ◽  
Extremal Problems ◽  
Dirac Type ◽  
Set Systems ◽  
Uniform Hypergraphs ◽  
Degree Conditions

We consider two extremal problems for set systems without long Berge cycles. First we give Dirac-type minimum degree conditions that force long Berge cycles. Next we give an upper bound for the number of hyperedges in a hypergraph with bounded circumference. Both results are best possible in infinitely many cases.

The proof of the upper bound in the multifractal formalism for chirps

2004 ◽  
Vol 130 (2) ◽  
pp. 99-110
Author(s):  
M BENSLIMANE
Keyword(s):  
Upper Bound ◽  
Multifractal Formalism

The spatial oscillation of velocity fields in the roll bite of strip cold rolling by the upper bound approach

2012 ◽  
Vol 109 (2) ◽  
pp. 73-80
Author(s):  
Q.-T. Ngo ◽  
A. Ehrlacher ◽  
N. Legrand
Keyword(s):  
Cold Rolling ◽  
Upper Bound ◽  
Velocity Fields ◽  
Spatial Oscillation ◽  
Roll Bite

Throughput Analyses Based on Practical Upper Bound for Adaptive Modulation and Coding in OFDM MIMO Multiplexing

2016 ◽  
Vol E99.A (1) ◽  
pp. 185-195
Author(s):  
Bing HAN ◽  
Teruo KAWAMURA ◽  
Yuichi KAKISHIMA ◽  
Mamoru SAWAHASHI
Keyword(s):  
Upper Bound ◽  
Adaptive Modulation ◽  
Adaptive Modulation And Coding

An Upper Bound on the Generalized Cayley Distance

2018 ◽  
Author(s):  
Akira Yamawaki ◽  
Hiroshi Kamabe ◽  
Shan Lu
Keyword(s):  
Upper Bound
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