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.

scholarly journals $S$-constrained random matrices

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Sylvain Gravier ◽  
Bernard Ycart
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Explicit Construction ◽  
Superimposed Codes ◽  
International Audience ◽  
Asymptotic Probability

International audience Let $S$ be a set of $d$-dimensional row vectors with entries in a $q$-ary alphabet. A matrix $M$ with entries in the same $q$-ary alphabet is $S$-constrained if every set of $d$ columns of $M$ contains as a submatrix a copy of the vectors in $S$, up to permutation. For a given set $S$ of $d$-dimensional vectors, we compute the asymptotic probability for a random matrix $M$ to be $S$-constrained, as the numbers of rows and columns both tend to infinity. If $n$ is the number of columns and $m=m_n$ the number of rows, then the threshold is at $m_n= \alpha_d \log (n)$, where $\alpha_d$ only depends on the dimension $d$ of vectors and not on the particular set $S$. Applications to superimposed codes, shattering classes of functions, and Sidon families of sets are proposed. For $d=2$, an explicit construction of a $S$-constrained matrix is given.

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 On the $L(p,1)$-labelling of graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Daniel Gonçalves
Keyword(s):  
Maximal Degree ◽  
International Audience

International audience In this paper we improve the best known bound for the $L(p,1)$-labelling of graphs with given maximal degree.

scholarly journals Quantitative and Algorithmic aspects of Barrier Synchronization in Concurrency

2021 ◽  
Vol vol. 22 no. 3, Computational... (Special issues) ◽  
Author(s):  
OLivier Bodini ◽  
Matthieu Dien ◽  
Antoine Genitrini ◽  
Frédéric Peschanski
Keyword(s):  
Integral Formula ◽  
Explicit Construction ◽  
Point Of View ◽  
Combinatorial Explosion ◽  
Control Graph ◽  
International Audience ◽  
Sampling Algorithms ◽  
Generate Process ◽  
Quantitative Results ◽  
Combinatorial Point

International audience In this paper we address the problem of understanding Concurrency Theory from a combinatorial point of view. We are interested in quantitative results and algorithmic tools to refine our understanding of the classical combinatorial explosion phenomenon arising in concurrency. This paper is essentially focusing on the the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space.

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].

scholarly journals Dyck path triangulations and extendability (extended abstract)

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Cesar Ceballos ◽  
Arnau Padrol ◽  
Camilo Sarmiento
Keyword(s):  
Cartesian Product ◽  
Explicit Construction ◽  
Oriented Matroids ◽  
Oriented Matroid ◽  
Dyck Path ◽  
Matroid Theory ◽  
Dyck Paths ◽  
Nous Montrons ◽  
International Audience

International audience We introduce the Dyck path triangulation of the cartesian product of two simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$ using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever$m\geq k>n$, any triangulations of $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique triangulation of $\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we prove that the bound $k>n$ is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory. Nous introduisons la triangulation par chemins de Dyck du produit cartésien de deux simplexes $\Delta_{n-1}\times\Delta_{n-1}$. Les simplexes maximaux de cette triangulation sont donnés par des chemins de Dyck, et cette construction se généralise de façon naturelle pour produire des triangulations $\Delta_{r\ n-1}\times\Delta_{n-1}$ qui utilisent des chemins de Dyck rationnels. Notre étude de la triangulation par chemins de Dyck est motivée par des problèmes de prolongement de triangulations partielles de produits de deux simplexes. On montre que $m\geq k>n$ alors toute triangulation de $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ se prolonge en une unique triangulation de $\Delta_{m-1}\times\Delta_{n-1}$. De plus, avec une construction explicite, nous montrons que la borne $k>n$ est optimale. Nous présentons aussi des interprétations de nos résultats dans le langage des matroïdes orientés tropicaux, qui sont analogues aux résultats classiques de la théorie des matroïdes orientés.

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 On the number of transversals in random trees

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Bernhard Gittenberger ◽  
Veronika Kraus
Keyword(s):  
Generating Functions ◽  
Singularity Analysis ◽  
Random Trees ◽  
Alternative Proof ◽  
Fixed Size ◽  
Random Subset ◽  
Polya Trees ◽  
Vertex Set ◽  
International Audience ◽  
Simply Generated Trees

International audience We study transversals in random trees with n vertices asymptotically as n tends to infinity. Our investigation treats the average number of transversals of fixed size, the size of a random transversal as well as the probability that a random subset of the vertex set of a tree is a transversal for the class of simply generated trees and for Pólya trees. The last parameter was already studied by Devroye for simply generated trees. We offer an alternative proof based on generating functions and singularity analysis and extend the result to Pólya trees.

scholarly journals A remark on goldbach's problem II

1984 ◽  
Vol Volume 7 ◽  
Author(s):  
S Srinivasan
Keyword(s):  
Alternative Proof ◽  
International Audience

International audience In this paper, we give an alternative proof of a theorem of R. Balasubramanian and C. J. Mozzochi

scholarly journals The purity of set-systems related to Grassmann necklaces

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Vladimir Danilov ◽  
Alexander Karzanov ◽  
Gleb Koshevoy
Keyword(s):  
Boolean Cube ◽  
Symmetric Relation ◽  
Prove Theorem ◽  
Set Systems ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets

International audience Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky introduced in 1998 the notion of weakly separated sets in $[n]:=\{1,\ldots, n\}$. Moreover, they raised several conjectures on the purity for this symmetric relation, in particular, on the Boolean cube $2^{[n]}$. In 0909.1423[math.CO] we proved these purity conjectures for the Boolean cube $2^{[n]}$, the discrete Grassmanian $\binom{[n]}{r}$, and some other set-systems. Oh, Postnikov, and Speyer in arxiv:1109.4434 proved the purity for weakly separated collections inside a positroid which contain a Grassmann necklace $\mathcal {N}$ defining the positroid. We denote such set-systems as $\mathcal{Int}(\mathcal {N} )$. In this paper we give an alternative (and shorter) proof of the purity of $\mathcal{Int}(\mathcal {N} )$ and present a stronger result. More precisely, we introduce a set-system $\mathcal{Out}(\mathcal {N} )$ complementary to $\mathcal{Int}(\mathcal {N })$, in a sense, and establish its purity. Moreover, we prove (Theorem~3) that these two set-systems are weakly separated from each other. As a consequence of Theorem~3, we obtain the purity of set-systems related to pairs of weakly separated necklaces (Proposition 4 and Corollaries 1 and 2). Finally, we raise a conjecture on the purity of both the interior and exterior of a generalized necklace.

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