Star Versus Two Stripes Ramsey Numbers and a Conjecture of Schelp

2012 ◽  
Vol 21 (1-2) ◽  
pp. 179-186 ◽  
Author(s):  
ANDRÁS GYÁRFÁS ◽  
GÁBOR N. SÁRKÖZY
Keyword(s):  
Minimum Degree ◽  
Ramsey Number ◽  
Regularity Lemma ◽  
Ramsey Numbers ◽  
Standard Application ◽  
Star Versus ◽  
Ramsey Type ◽  
Weakened Form

R. H. Schelp conjectured that if G is a graph with |V(G)| = R(Pn, Pn) such that δ(G) > $$\frac{3|V(G)|}{ 4}$, then in every 2-colouring of the edges of G there is a monochromatic Pn. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree.Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching–matching Ramsey number satisfying R(Sn, nK2, nK2) = 3n − 1. This extends R(nK2, nK2) = 3n − 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma.It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.

scholarly journals On the Size-Ramsey Number of Cycles

2019 ◽  
Vol 28 (06) ◽  
pp. 871-880
Author(s):  
R. Javadi ◽  
F. Khoeini ◽  
G. R. Omidi ◽  
A. Pokrovskiy
Keyword(s):  
Upper Bounds ◽  
Ramsey Number ◽  
Alternative Proof ◽  
Regularity Lemma ◽  
Ramsey Numbers ◽  
Szemeredi's Regularity Lemma ◽  
Number Of Cycles ◽  
Image Position ◽  
Edge Colouring ◽  
Monochromatic Copy

AbstractFor given graphs G1,…, Gk, the size-Ramsey number $\hat R({G_1}, \ldots ,{G_k})$ is the smallest integer m for which there exists a graph H on m edges such that in every k-edge colouring of H with colours 1,…,k, H contains a monochromatic copy of Gi of colour i for some 1 ≤ i ≤ k. We denote $\hat R({G_1}, \ldots ,{G_k})$ by ${\hat R_k}(G)$ when G1 = ⋯ = Gk = G.Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle Cn is linear in n, ${\hat R_k}({C_n}) \le {c_k}n$ for some constant ck. Their proof, however, is based on Szemerédi’s regularity lemma so no specific constant ck is known.In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of ${\hat R_k}({C_n}) \le {c_k}n$ , avoiding use of the regularity lemma, where ck is exponential and doubly exponential in k, when n is even and odd, respectively. In particular, we show that for sufficiently large n we have ${\hat R_2}({C_n}) \le {10^5} \times cn$ , where c = 6.5 if n is even and c = 1989 otherwise.

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 Batas Bawah Bilangan Ramsey pada Graf Bintang S_10 Versus Roda W_12 Lower Bound of the Ramsey Number for the Star Graph S_10 Versus Wheels W_12

2018 ◽  
Vol 3 (1) ◽  
pp. 471
Author(s):  
Hamdana Hadaming ◽  
Andi Ardhila Wahyudi
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lower Boundary ◽  
Ramsey Number ◽  
Star Graph ◽  
Point Graph ◽  
Ramsey Numbers ◽  
Critical Graph ◽  
Star Versus

Bilangan Ramsey untuk graf  terhadap graf , dinotasikan dengan  adalah bilangan bulat terkecil  sedemikian sehingga untuk setiap graf  dengan orde akan memenuhi sifat berikut:  memuat graf  atau komplemen dari  memuat graf .Penelitian ini bertujuan untuk  menentukan graf kritis maksimum  dan  dengan genap. Berdasarkan batas bawah tersebut di tentukan batas atas minimum sehingga diperoleh nilai bilangan Ramsey untuk graf bintang  versus , atau . Dengan demikian penentuan batas bawah bilangan Ramsey  dilakukan dengan cara batas bawah yang  diberikan oleh Chavatal dan Harary, untuk bilangan Ramsey pada graf bintang  versus  adalah , dengan  adalah bilangan kromatik titik graf roda  dan  adalah kardinalitas komponen terbesar graf . Berdasarkan batas bawah Chavatal dan Harary tersebut dikonstruksi graf kritis untuk  dan  yang ordenya lebih besar dari nilai batas bawah yang diberikan Chavatal dan Harary. Orde dari graf kritis tersebut merupakan batas bawah terbaik untuk . Kata kunci: Bilangan Ramsey, bintang, roda AbstractRamsey Numbers for a graph  to a graph , denoted by   is the smallest integer n such that for every graph  of order  either  the following meeet:  contains a graph  or the complement of  contains the graph . This aims of the study to determine the maximum critical graph  and . Based on the lower bound of the specified minimum upper bound in order to obtain numerical values for the Ramsey graph  Star versus  , or . Thus the determination of Ramsey numbers .  is done by determine the lower boundary and upper bound. The lower bound given by Chavatal and Harary, for ramsey number for star graph versus wheel   is , is a point graph of chromatic number wheel  and  is the cardinality of the largest component of the graph . Based on the lower bound Chavatal and Harary graph is constructed critical to  and  are poin greater than the lower bound value given Chavatal and Harary. Order of the critical graph is the best lower bound for . Keywords : Ramsey number, Stars, and Wheels

scholarly journals Multicolor Ramsey Numbers and Restricted Turán Numbers for the Loose 3-Uniform Path of Length Three

10.37236/7119 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Andrzej Ruciński ◽  
Eliza Jackowska ◽  
Joanna Polcyn
Keyword(s):  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Ramsey Numbers ◽  
Turán Number ◽  
Standard Application

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Turán number $\mathrm{ex}_3(n;P)$, which was determined to be $\binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Turán numbers for $P$ under some additional restrictions. Most importantly, we determine the largest number of edges in an $n$-vertex $P$-free 3-graph which is not a star. These Turán-type results, in turn, allow us to confirm the formula $R(P;r)=r+6$ for $r\in\{4,5,6,7\}$.

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.

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