scholarly journals The Ramsey Number of Loose Paths in 3-Uniform Hypergraphs

10.37236/2725 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Leila Maherani ◽  
Gholam Reza Omidi ◽  
Ghaffar Raeisi ◽  
Maryam Shahsiah
Keyword(s):  
The Other ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Uniform Hypergraphs ◽  
Asymptotic Values

Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of $3$-uniform loose paths when one of the paths is significantly larger than the other:  for every $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$, we show that $$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor.$$

The 3-Colour Ramsey Number of a 3-Uniform Berge Cycle

2010 ◽  
Vol 20 (1) ◽  
pp. 53-71 ◽  
Author(s):  
ANDRÁS GYÁRFÁS ◽  
GÁBOR N. SÁRKÖZY
Keyword(s):  
Ramsey Number ◽  
Regularity Lemma ◽  
Ramsey Numbers ◽  
Uniform Hypergraphs

The asymptotics of 2-colour Ramsey numbers of loose and tight cycles in 3-uniform hypergraphs were recently determined [16, 17]. We address the same problem for Berge cycles and for 3 colours. Our main result is that the 3-colour Ramsey number of a 3-uniform Berge cycle of length n is asymptotic to $\frac{5n}{4}$. The result is proved with the Regularity Lemma via the existence of a monochromatic connected matching covering asymptotically 4n/5 vertices in the multicoloured 2-shadow graph induced by the colouring of Kn(3).

scholarly journals The Ramsey Number of Loose Triangles and Quadrangles in Hypergraphs

10.37236/2346 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Andras Gyarfas ◽  
Ghaffar Raeisi
Keyword(s):  
Ramsey Number ◽  
Ramsey Numbers ◽  
Asymptotic Values

Asymptotic values of hypergraph Ramsey numbers for loose cycles (and paths) were determined recently. Here we determine some of them exactly, for example the 2-color hypergraph Ramsey number of a $k$-uniform loose 3-cycle or 4-cycle: $R(\mathcal{C}^k_3,\mathcal{C}^k_3)=3k-2$ and $R(\mathcal{C}_4^k,\mathcal{C}_4^k)=4k-3$ (for $k\geq 3$). For more than 3-colors we could prove only that $R(\mathcal{C}^3_3,\mathcal{C}^3_3,\mathcal{C}^3_3)=8$. Nevertheless, the $r$-color Ramsey number of triangles for hypergraphs are much smaller than for graphs: for $r\geq 3$, $$r+5\le R(\mathcal{C}_3^3,\mathcal{C}_3^3,\dots,\mathcal{C}_3^3)\le 3r$$

scholarly journals A Note on On-Line Ramsey Numbers for Some Paths

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 735
Author(s):  
Tomasz Dzido ◽  
Renata Zakrzewska
Keyword(s):  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
On Line ◽  
Infinite Set ◽  
Do So

We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.

scholarly journals Ramsey Numbers for Theta Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
M. M. M. Jaradat ◽  
M. S. A. Bataineh ◽  
S. M. E. Radaideh
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Subgraph Isomorphic

The graph Ramsey number is the smallest integer with the property that any complete graph of at least vertices whose edges are colored with two colors (say, red and blue) contains either a subgraph isomorphic to all of whose edges are red or a subgraph isomorphic to all of whose edges are blue. In this paper, we consider the Ramsey numbers for theta graphs. We determine , for . More specifically, we establish that for . Furthermore, we determine for . In fact, we establish that if is even, if is odd.

scholarly journals The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 764
Author(s):  
Yaser Rowshan ◽  
Mostafa Gholami ◽  
Stanford Shateyi
Keyword(s):  
Positive Integer ◽  
Ramsey Number ◽  
Complete Multipartite Graph ◽  
Ramsey Numbers ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Monochromatic Copy

For given graphs G1,G2,…,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,…,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n≥2 and mj(nK2,C7), for any j≤4 and n≥2.

Mixed Ramsey Numbers Revisited

2003 ◽  
Vol 12 (5-6) ◽  
pp. 653-660
Author(s):  
C. C. Rousseau ◽  
S. E. Speed
Keyword(s):  
Asymptotic Behaviour ◽  
Ramsey Number ◽  
Open Problems ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Dense Graphs

Given a graph Hwith no isolates, the (generalized) mixed Ramsey number is the smallest integer r such that every H-free graph of order r contains an m-element irredundant set. We consider some questions concerning the asymptotic behaviour of this number (i) with H fixed and , (ii) with m fixed and a sequence of dense graphs, in particular for the sequence . Open problems are mentioned throughout the paper.

scholarly journals Minimum Clique Number, Chromatic Number, and Ramsey Numbers

10.37236/2125 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Chromatic Number ◽  
Asymptotic Properties ◽  
Inverse Function ◽  
Clique Number ◽  
Ramsey Number ◽  
Growth Order ◽  
Ramsey Numbers

Let $Q(n,c)$ denote the minimum clique number over graphs with $n$ vertices and chromatic number $c$. We investigate the asymptotics of $Q(n,c)$ when $n/c$ is held constant. We show that when $n/c$ is an integer $\alpha$, $Q(n,c)$ has the same growth order as the inverse function of the Ramsey number $R(\alpha+1,t)$ (as a function of $t$). Furthermore, we show that if certain asymptotic properties of the Ramsey numbers hold, then $Q(n,c)$ is in fact asymptotically equivalent to the aforementioned inverse function. We use this fact to deduce that $Q(n,\lceil n/3 \rceil)$ is asymptotically equivalent to the inverse function of $R(4,t)$.

scholarly journals Constrained Ramsey Numbers

2009 ◽  
Vol 18 (1-2) ◽  
pp. 247-258 ◽  
Author(s):  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Ramsey Theorem ◽  
Logarithmic Factor ◽  
Subgraph Isomorphic ◽  
Rate Of Growth ◽  
Special Cases ◽  
Edge Colouring

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

scholarly journals COMPUTATION OF RAMSEY NUMBERS BY P SYSTEMS WITH ACTIVE MEMBRANES

2011 ◽  
Vol 22 (01) ◽  
pp. 29-38 ◽  
Author(s):  
LINQIANG PAN ◽  
DANIEL DÍAZ-PERNIL ◽  
MARIO J. PÉREZ-JIMÉNEZ
Keyword(s):  
Membrane Computing ◽  
P Systems ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Combinatorial Object ◽  
Active Membranes ◽  
Number Problem ◽  
Decision Version

Ramsey numbers deal with conditions when a combinatorial object necessarily contains some smaller given objects. It is well known that it is very difficult to obtain the values of Ramsey numbers. In this work, a theoretical chemical/biological solution is presented in terms of membrane computing for the decision version of Ramsey number problem, that is, to decide whether an integer n is the value of Ramsey number R(k, l), where k and l are integers.

scholarly journals HYPERGRAPH RAMSEY NUMBERS AND ADIABATIC QUANTUM ALGORITHM

2012 ◽  
Vol 10 (06) ◽  
pp. 1250067 ◽  
Author(s):  
RI QU ◽  
SU-LI ZHAO ◽  
YAN-RU BAO ◽  
XIAO-CHUN CAO
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
Quantum Algorithm ◽  
Combinatorial Optimization Problem ◽  
Quantum Evolution ◽  
Ramsey Numbers ◽  
Uniform Hypergraphs

Gaitan and Clark [Phys. Rev. Lett.108 (2012) 010501] have recently presented a quantum algorithm for the computation of the Ramsey numbers R(m, n) using adiabatic quantum evolution (AQE). We consider that the two-color Ramsey numbers R(m, n; r) for r-uniform hypergraphs can be computed by using the similar ways in Phys. Rev. Lett.108 (2012) 010501. In this paper, we show how the computation of R(m, n; r) can be mapped to a combinatorial optimization problem whose solution be found using AQE.

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