scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals Equations with powers of singular moduli

2019 ◽  
Vol 15 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Antonin Riffaut
Keyword(s):  
Rational Number ◽  
The Other ◽  
Other Hand ◽  
Singular Moduli ◽  
Cm Points ◽  
The One ◽  
Straight Lines ◽  
Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

Arithmetic functions monotonic at consecutive arguments

2014 ◽  
Vol 51 (2) ◽  
pp. 155-164
Author(s):  
Jean-Marie Koninck ◽  
Florian Luca
Keyword(s):  
Large Class ◽  
The Other ◽  
Arithmetic Functions ◽  
Arbitrary Integer ◽  
Other Hand ◽  
Positive Integers

For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|nd2. On the other hand, we prove that for the function f(n) := ∑p|np2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.

scholarly journals Upper density of monochromatic infinite paths

2019 ◽  
Author(s):  
Jan Corsten ◽  
Louis DeBiasio ◽  
Ander Lamaison ◽  
Richard Lang
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Classical Case ◽  
Infinite Graphs ◽  
Edge Colorings ◽  
Vertex Set ◽  
Upper Density ◽  
Countably Infinite ◽  
Positive Integers ◽  
Red Path

Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.

Two Extremal Problems in Graph Theory

10.37236/1182 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Richard A. Brualdi ◽  
Stephen Mellendorf
Keyword(s):  
Graph Theory ◽  
Extremal Problems ◽  
Turan's Theorem ◽  
Turán’S Theorem ◽  
Positive Integers

We consider the following two problems. (1) Let $t$ and $n$ be positive integers with $n\geq t\geq 2$. Determine the maximum number of edges of a graph of order $n$ that contains neither $K_t$ nor $K_{t,t}$ as a subgraph. (2) Let $r$, $t$ and $n$ be positive integers with $n\geq rt$ and $t\geq 2$. Determine the maximum number of edges of a graph of order $n$ that does not contain $r$ disjoint copies of $K_t$. Problem 1 for $n < 2t$ is solved by Turán's theorem and we solve it for $n=2t$. We also solve Problem 2 for $n=rt$.

On generalised Fibonaccian and Lucasian numbers

2006 ◽  
Vol 90 (518) ◽  
pp. 215-222 ◽  
Author(s):  
Peter Hilton ◽  
Jean Pedersen
Keyword(s):  
Residue Class ◽  
The Other ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Residue Classes ◽  
Other Hand ◽  
Positive Integers ◽  
Striking Fact

In [1, Chapter 3, Section 2], we collected together results we had previously obtained relating to the question of which positive integers m were Lucasian, that is, factors of some Lucas number L n. We pointed out that the behaviors of odd and even numbers m were quite different. Thus, for example, 2 and 4 are both Lucasian but 8 is not; for the sequence of residue classes mod 8 of the Lucas numbers, n ⩾ 0, reads and thus does not contain the residue class 0*. On the other hand, it is a striking fact that, if the odd number s is Lucasian, then so are all of its positive powers.

scholarly journals On the size of Diophantine m-tuples

2002 ◽  
Vol 132 (1) ◽  
pp. 23-33 ◽  
Author(s):  
ANDREJ DUJELLA
Keyword(s):  
The Other ◽  
Famous Conjecture ◽  
Nonzero Integer ◽  
Other Hand ◽  
Parametric Families ◽  
Diophantine Quintuples ◽  
Positive Integers

Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D(1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [12, 14, 16], but the conjecture is still unproved.On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920, 32768, 44460, 19534284} with the property D(2985984) [19]].

scholarly journals Discrepancy of Matrices of Zeros and Ones

10.37236/1447 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
R. A. Brualdi ◽  
J. Shen
Keyword(s):  
Explicit Formula ◽  
The Other ◽  
Column Vector ◽  
Other Hand ◽  
Minimum Discrepancy ◽  
Positive Integers ◽  
Maximum Discrepancy ◽  
Sum Vector

Let $m$ and $n$ be positive integers, and let $R=(r_1,\ldots, r_m)$ and $ S=(s_1,\ldots, s_n)$ be non-negative integral vectors. Let ${\cal A} (R,S)$ be the set of all $m \times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$, and let $\bar A$ be the $m \times n$ $(0,1)$-matrix where for each $i$, $1\le i \le m$, row $i$ consists of $r_i$ $1$'s followed by $n-r_i$ $0$'s. If $S$ is monotone, the discrepancy $d(A)$ of $A$ is the number of positions in which $\bar A$ has a $1$ and $A$ has a $0$. It equals the number of $1$'s in $\bar A$ which have to be shifted in rows to obtain $A$. In this paper, we study the minimum and maximum $d(A)$ among all matrices $A \in {\cal A} (R,S)$. We completely solve the minimum discrepancy problem by giving an explicit formula in terms of $R$ and $S$ for it. On the other hand, the problem of finding an explicit formula for the maximum discrepancy turns out to be very difficult. Instead, we find an algorithm to compute the maximum discrepancy.

scholarly journals Dynamics of Boundary Graphs

2013 ◽  
Vol 5 (3) ◽  
pp. 447-455
Author(s):  
G. Mariumuthu ◽  
M. S. Saraswathy
Keyword(s):  
Simple Graph ◽  
The Other ◽  
Boundary Vertex ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Periodic Graph ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Boundary Graph

In a graph G, the distance d(u,v) between a pair of vertices u and v is the length of a shortest path joining them. A vertex v is a boundary vertex of a vertex u if for all The boundary graph B(G) based on a connected graph G is a simple graph which has the vertex set as in G. Two vertices u and v are adjacent in B(G) if either u is a boundary of v or v is a boundary of u. If G is disconnected, then each vertex in a component is adjacent to all other vertices in the other components and is adjacent to all of its boundary vertices within the component. Given a positive integer m, the mth iterated boundary graph of G is defined as A graph G is periodic if for some m. A graph G is said to be an eventually periodic graph if there exist positive integers m and k >0 such that We give the necessary and sufficient condition for a graph to be eventually periodic.  Keywords: Boundary graph; Periodic graph. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v5i3.14866 J. Sci. Res. 5 (3), xxx-xxx (2013) 

scholarly journals $k$-Fold Sidon Sets

10.37236/3860 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Javier Cilleruelo ◽  
Craig Timmons
Keyword(s):  
The Other ◽  
Sidon Sets ◽  
Sidon Set ◽  
Other Hand ◽  
Positive Integers

Let $k \geq 1$ be an integer.  A set $A \subset \mathbb{Z}$ is a $k$-fold Sidon set if $A$ has only trivial solutions to each equation of the form $c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ where $0 \leq |c_i | \leq k$, and $c_1 + c_2 + c_3 + c_4 = 0$.  We prove that for any integer $k \geq 1$, a $k$-fold Sidon set $A \subset [N]$ has at most $(N/k)^{1/2} + O((Nk)^{1/4})$ elements. Indeed we prove that given any $k$ positive integers $c_1<\cdots <c_k$, any set $A\subset [N]$ that contains only trivial solutions to $c_i(x_1-x_2)=c_j(x_3-x_4)$ for each $1 \le i \le j \le k$, has at most $(N/k)^{1/2}+O((c_k^2N/k)^{1/4})$ elements. On the other hand, for any $k \geq 2$ we can exhibit $k$ positive integers $c_1,\dots, c_k$ and a set $A\subset [N]$ with $|A|\ge (\frac 1k+o(1))N^{1/2}$, such that $A$ has only trivial solutions to $c_i(x_1 - x_2) = c_j (x_3 -  x_4)$ for each $1 \le i \le j\le k$.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

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