scholarly journals On zero-sum turan problems of Bialostocki and Dierker

1992 ◽  
Vol 53 (3) ◽  
pp. 402-407 ◽  
Author(s):  
Y. Caro ◽  
Y. Roditty
Keyword(s):  
Additive Group ◽  
The Family ◽  
Turán Number ◽  
Turan Problems ◽  
Zero Sum

AbstractAssume G is a graph with m edges. By T(n, G) we denote the classical Turan number, namely, the maximum possible number of edges in a graph H on n vertices without a copy of G. Similarly if G is a family of graphs then H does not have a copy of any member of the family. A Zk-colouring of a graph G is a colouring of the edges of G by Zk, the additive group of integers modulo k, avoiding a copy of a given graph H, for which the sum of the values on its edges is 0 (mod k). By the Zero-Sum Turan number, denoted T(n, G, Zk), k¦m, we mean the maximum number of edges in a Zk-colouring of a graph on n vertices that contains no zero-sum (mod k) copy of G. Here we mainly solve two problems of Bialostocki and Dierker [6].Problem 1. Determine T(n, tK2, Zk) for ¦|t. In particular, is it true that T(n, tK2, Zk) = T(n, (t+k-1)K2)?Problem 2. Does there exist a constant c(t, k) such that T(n, Ft, Zk) ≦ c(t, k)n, where Ft is the family of cycles of length at least t?

scholarly journals An Ordered Turán Problem for Bipartite Graphs

10.37236/2471 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Craig Timmons
Keyword(s):  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Free Graph ◽  
The Family ◽  
Turan Problem ◽  
Vertex Graph ◽  
Turán Number ◽  
Turán Problem ◽  
Natural Way

Let $F$ be a graph.  A graph $G$ is $F$-free if it does not contain $F$ as a subgraph.  The Turán number of $F$, written $\textrm{ex}(n,F)$, is the maximum number of edges in an $F$-free graph with $n$ vertices.  The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem.  In this paper we introduce an ordered version of the Turán problem for bipartite graphs.  Let $G$ be a graph with $V(G) = \{1, 2, \dots , n \}$ and view the vertices of $G$ as being ordered in the natural way.  A zig-zag $K_{s,t}$, denoted $Z_{s,t}$, is a complete bipartite graph $K_{s,t}$ whose parts $A = \{n_1 < n_2 < \dots < n_s \}$ and $B = \{m_1 < m_2 < \dots < m_t \}$ satisfy the condition $n_s < m_1$.  A zig-zag $C_{2k}$ is an even cycle $C_{2k}$ whose vertices in one part precede all of those in the other part.  Write $\mathcal{Z}_{2k}$ for the family of zig-zag $2k$-cycles.  We investigate the Turán numbers $\textrm{ex}(n,Z_{s,t})$ and $\textrm{ex}(n,\mathcal{Z}_{2k})$.  In particular we show $\textrm{ex}(n, Z_{2,2}) \leq \frac{2}{3}n^{3/2} + O(n^{5/4})$.  For infinitely many $n$ we construct a $Z_{2,2}$-free $n$-vertex graph with more than $(n - \sqrt{n} - 1) + \textrm{ex} (n,K_{2,2})$ edges.

scholarly journals On the Chromatic Thresholds of Hypergraphs

2015 ◽  
Vol 25 (2) ◽  
pp. 172-212
Author(s):  
JÓZSEF BALOGH ◽  
JANE BUTTERFIELD ◽  
PING HU ◽  
JOHN LENZ ◽  
DHRUV MUBAYI
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Open Problems ◽  
Fibre Bundles ◽  
Uniform Hypergraphs ◽  
The Family ◽  
Turán Number ◽  
Chromatic Thresholds ◽  
Extremal Set

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

scholarly journals Rainbow Turán Problems for Paths and Forests of Stars

10.37236/6430 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Daniel Johnston ◽  
Cory Palmer ◽  
Amites Sarkar
Keyword(s):  
Systematic Study ◽  
Colored Graph ◽  
Turán Number ◽  
Turan Problems ◽  
Different Color

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the rainbow Turán number of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte [Combinatorics, Probability and Computing 16 (2007)]. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $l$ edges, disproving a conjecture in the aforementioned paper for $l=4$.

scholarly journals Metric spaces related to Abelian groups

2021 ◽  
Vol 22 (1) ◽  
pp. 169
Author(s):  
Amir Veisi ◽  
Ali Delbaznasab
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Additive Group ◽  
Ordered Group ◽  
Infinite Cyclic Group ◽  
Ordered Groups ◽  
The Family ◽  
Nonempty Compact ◽  
Metric Topology ◽  
Induced Topology

<p>When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ cl<sub>X</sub>A if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e &lt; g ∈ G and d is bounded, then d′ (A, B) &lt; g if and only if A ⊆ N<sub>d</sub>(B, g) and B ⊆ N<sub>d</sub>(A, g), where N<sub>d</sub>(A, g) = {x ∈ X : d(x, A) &lt; g}, d<sub>B</sub>(A) = sup{d(a, B) : a ∈ A} and d′ (A, B) = sup{d<sub>A</sub>(B), d<sub>B</sub>(A)}.</p>

scholarly journals Some Exact Results and New Asymptotics for Hypergraph Turán Numbers

2002 ◽  
Vol 11 (3) ◽  
pp. 299-309 ◽  
Author(s):  
DHRUV MUBAYI
Keyword(s):  
Lower Bound ◽  
Explicit Construction ◽  
Bipartite Graphs ◽  
Probabilistic Methods ◽  
Exact Results ◽  
Open Problems ◽  
Asymptotic Value ◽  
The Family ◽  
Turán Number

Given a family [Fscr ] of r-graphs, let ex(n, [Fscr ]) be the maximum number of edges in an n-vertex r-graph containing no member of [Fscr ]. Let C(r)4 denote the family of r-graphs with distinct edges A, B, C, D, such that A ∩ B = C ∩ D = Ø and A ∪ B = C ∪ D. For s1 [les ] … [les ] sr, let K(r) (s1,…,sr) be the complete r-partite r-graph with parts of sizes s1,…,sr.Füredi conjectured over 15 years ago that ex(n,C(3)4) [les ] (n2) for sufficiently large n. We prove the weaker resultex(n, {C(3)4, K(3)(1,2,4)}) [les ] (n2).Generalizing a well-known conjecture for the Turán number of bipartite graphs, we conjecture thatex(n, K(r)(s1,…,sr)) = Θ(nr−1/s),where s = Πr−1i=1si. We prove this conjecture when s1 = … = sr−2 = 1 and(i) sr−1 = 2, (ii) sr−1 = sr = 3, (iii)sr > (sr−1−1)!.In cases (i) and (ii), we determine the asymptotic value of ex(n,K(r)(s1,…,sr)).We also provide an explicit construction givingex(n,K(3)(2,2,3)) > (1/6−o(1))n8/3.This improves upon the previous best lower bound of Ω(n29/11) obtained by probabilistic methods. Several related open problems are also presented.

Varieties of topological groups III

1970 ◽  
Vol 2 (2) ◽  
pp. 165-178 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Algebraic Variety ◽  
Additive Group ◽  
Topological Groups ◽  
Locally Compact ◽  
Circle Group ◽  
Free Topological Group ◽  
The Family ◽  
Usual Topology

This paper continues the invèstigation of varieties of topological groups. It is shown that the family of all varieties of topological groups with any given underlying algebraic variety is a class and not a set. In fact the family of all β-varieties with any given underlying algebraic variety is a class and not a set. A variety generated by a family of topological groups of bounded cardinal is not a full variety.The varieties V(R) and V(T) generated by the additive group of reals and the circle group respectively each with its usual topology are examined. In particular it is shown that a locally compact Hausdorff abelian group is in V(T) if and only if it is compact. Thus V(R) properly contains V(T).It is proved that any free topological group of a non-indiscrete variety is disconnected. Finally, some comments are made on topologies on free groups.

scholarly journals The construction of national identities

2020 ◽  
Vol 15 (2) ◽  
pp. 763-810 ◽  
Author(s):  
Milena Almagro ◽  
David Andrés-Cerezo
Keyword(s):  
Central Government ◽  
Initial Conditions ◽  
Nation Building ◽  
National Identities ◽  
Historical Factors ◽  
Long Run ◽  
The Family ◽  
Zero Sum ◽  
State Centralization ◽  
Country Specific

This paper explores the dynamics of nation‐building policies and the conditions under which a state can promote a shared national identity on its territory. A forward‐looking central government that internalizes identity dynamics shapes them by choosing the level of state centralization. Homogenization attempts are constrained by political unrest, electoral competition and the intergenerational transmission of identities within the family. We find nation‐building efforts are generally characterized by fast interventions. We show that a zero‐sum conflict over resources pushes long‐run dynamics toward homogeneous steady states and extreme levels of (de)centralization. We also find the ability to foster a common identity is highly dependent on initial conditions, and that country‐specific historical factors can have a lasting impact on the long‐run distribution of identities.

scholarly journals Actions of the additive group Ga on certain noncommutative deformations of the plane

2021 ◽  
Vol 29 (2) ◽  
pp. 269-279
Author(s):  
Ivan Kaygorodov ◽  
Samuel A. Lopes ◽  
Farukh Mashurov
Keyword(s):  
Automorphism Group ◽  
Polynomial Ring ◽  
Characteristic Zero ◽  
Weyl Algebra ◽  
Additive Group ◽  
Prime Characteristic ◽  
Arbitrary Characteristic ◽  
Arbitrary Polynomial ◽  
Comodule Algebra ◽  
The Family

Abstract We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras A h = 〈 x , y | y x − x y = h ( x ) 〉 , {A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle , , where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah . We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah .

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