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 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 $t$-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 Optimal and Nonoptimal Gronwall Lemmas

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1728
Author(s):  
Daniela Marian ◽  
Sorina Anamaria Ciplea ◽  
Nicolaie Lungu
Keyword(s):  
Upper Bound ◽  
Partial Answer ◽  
Optimal Inequalities

In this paper, we study some optimal inequalities of the Riccati type and of the Bihari type. We also consider nonoptimal inequalities of the Wendorf type. At the same time, we get a partial answer to Problems 5 and 9, formulated by I. A. Rus. This paper is also motivated by the fact that, in many inequalities, the upper bound is not an optimal one.

scholarly journals On the density and the structure of the Peirce-like formulae

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Antoine Genitrini ◽  
Jakub Kozik ◽  
Grzegorz Matecki
Keyword(s):  
Finite Number ◽  
Upper Bound ◽  
Large Family ◽  
Partial Answer ◽  
International Audience ◽  
Almost All

International audience Within the language of propositional formulae built on implication and a finite number of variables $k$, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae - Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different $k$ is asymptotically equivalent to the sequence $\frac{1}{ 2 k^2}$. We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by $\frac{c}{ k^3}$ for some constant $c \in \mathbb{R}$. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc - although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to $\frac{1}{ 2 k^2}$.

scholarly journals Multivariate Lagrange Interpolation at Sinc Points Error Estimation and Lebesgue Constant

2016 ◽  
Vol 8 (4) ◽  
pp. 118 ◽  
Author(s):  
Maha Youssef ◽  
Hany A. El-Sharkawy ◽  
Gerd Baumann
Keyword(s):  
Uniform Convergence ◽  
Error Estimation ◽  
Upper Bound ◽  
Lagrange Interpolation ◽  
Lebesgue Constant ◽  
Explicit Construction ◽  
Interpolation Operator ◽  
Lebesgue Constants ◽  
Set Of Points

This paper gives an explicit construction of multivariate Lagrange interpolation at Sinc points. A nested operator formula for Lagrange interpolation over an $m$-dimensional region is introduced. For the nested Lagrange interpolation, a proof of the upper bound of the error is given showing that the error has an exponentially decaying behavior. For the uniform convergence the growth of the associated norms of the interpolation operator, i.e., the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature $O((log n)^m)$. We compare the obtained Lebesgue constant bound with other well known bounds for Lebesgue constants using different set of points.

scholarly journals B.H. Neumann's Question on Ensuring Commutativity of Finite Groups

2006 ◽  
Vol 74 (1) ◽  
pp. 121-132 ◽  
Author(s):  
A. Abdollahi ◽  
A. Azad ◽  
A. Mohammadi Hassanabadi ◽  
M. Zarrin
Keyword(s):  
Finite Group ◽  
Exponential Function ◽  
Finite Groups ◽  
Upper Bound ◽  
Partial Answer ◽  
A Finite Group

This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.

scholarly journals A Note on Set Systems with no Union of Cardinality 0 modulo m

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Vince Grolmusz
Keyword(s):  
Prime Power ◽  
Explicit Construction ◽  
Alternative Proof ◽  
Maximal Degree ◽  
Set Systems ◽  
International Audience

International audience \emphAlon, Kleitman, Lipton, Meshulam, Rabin and \emphSpencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph \textbf\textitF=\F_1,F_2,\ldots, F_d(q-1)+1\, where q is a prime-power, and d denotes the maximal degree of the hypergraph, there exists an \textbf\textitF_0⊂ \textbf\textitF, such that |\bigcup_F∈\textbf\textitF_0F| ≡ 0 (q). We give a direct, alternative proof for this theorem, and we also show that an explicit construction exists for a hypergraph of degree d and size Ω (d^2) which does not contain a non-empty sub-hypergraph with a union of size 0 modulo 6, consequently, the theorem does not generalize for non-prime-power moduli.

State Complexity of k-Union and k-Intersection for Prefix-Free Regular Languages

2015 ◽  
Vol 26 (02) ◽  
pp. 211-227 ◽  
Author(s):  
Hae-Sung Eom ◽  
Yo-Sub Han ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Finite Automata ◽  
Constant Factor ◽  
The State ◽  
Regular Languages ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Binary Alphabet ◽  
Multiple Intersections

We investigate the state complexity of multiple unions and of multiple intersections for prefix-free regular languages. Prefix-free deterministic finite automata have their own unique structural properties that are crucial for obtaining state complexity upper bounds that are improved from those for general regular languages. We present a tight lower bound construction for k-union using an alphabet of size k + 1 and for k-intersection using a binary alphabet. We prove that the state complexity upper bound for k-union cannot be reached by languages over an alphabet with less than k symbols. We also give a lower bound construction for k-union using a binary alphabet that is within a constant factor of the upper bound.

ON WIDE DIAMETER OF CAYLEY GRAPHS

2009 ◽  
Vol 10 (03) ◽  
pp. 219-231 ◽  
Author(s):  
KEVIN E. JIA ◽  
ALICIA Q. ZHANG
Keyword(s):  
Cayley Graph ◽  
Cyclic Group ◽  
Open Problem ◽  
Network Theory ◽  
Cayley Graphs ◽  
Wide Diameter ◽  
Residue Classes ◽  
Element Set

Because of its applications in network theory, the wide diameter of graphs has been studied extensively in recent years. It is still an open problem to compute the wide diameter of Cayley graphs. This paper investigates the wide diameter of Cayley graphs of ℤm, the cyclic group of residue classes modulo m. It is proved that the k-wide diameter of the Cayley graph Cay (Zm, A) generated by a k-element set A is d + 1 for k = 2 and ≤ d + 1 for k = 3, where d is the diameter of Cay (ℤm, A).

scholarly journals Constructing Large Set Systems with Given Intersection Sizes Modulo Composite Numbers

2002 ◽  
Vol 11 (5) ◽  
pp. 475-486 ◽  
Author(s):  
SAMUEL KUTIN
Keyword(s):  
Prime Power ◽  
Large Set ◽  
Residue Classes ◽  
Set Systems ◽  
Pairwise Intersection ◽  
Technical Report ◽  
Original Argument

We consider k-uniform set systems over a universe of size n such that the size of each pairwise intersection of sets lies in one of s residue classes mod q, but k does not lie in any of these s classes. A celebrated theorem of Frankl and Wilson [8] states that any such set system has size at most (ns) when q is prime. In a remarkable recent paper, Grolmusz [9] constructed set systems of superpolynomial size Ω(exp(c log2n/log log n)) when q = 6. We give a new, simpler construction achieving a slightly improved bound. Our construction combines a technique of Frankl [6] of ‘applying polynomials to set systems’ with Grolmusz's idea of employing polynomials introduced by Barrington, Beigel and Rudich [5]. We also extend Frankl's original argument to arbitrary prime-power moduli: for any ε > 0, we construct systems of size ns+g(s), where g(s) = Ω(s1−ε). Our work overlaps with a very recent technical report by Grolmusz [10].

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