Note on the Multicolour Size-Ramsey Number for Paths,

The Ramsey number $r(C_l, K_n)$ is the smallest positive integer $m$ such that every graph of order $m$ contains either cycle of length $l$ or a set of $n$ independent vertices. In this short note we slightly improve the best known upper bound on $r(C_l, K_n)$ for odd $l$.

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On the Size-Ramsey Number of Cycles

Combinatorics Probability Computing ◽

10.1017/s0963548319000221 ◽

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.

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An Alternative Proof of the Linearity of the Size-Ramsey Number of Paths

Combinatorics Probability Computing ◽

10.1017/s096354831400056x ◽

2014 ◽

Vol 24 (3) ◽

pp. 551-555 ◽

Cited By ~ 16

Author(s):

ANDRZEJ DUDEK ◽

PAWEŁ PRAŁAT

Keyword(s):

Ramsey Number ◽

Alternative Proof ◽

Monochromatic Copy

The size-Ramsey number $\^{r} $(F) of a graph F is the smallest integer m such that there exists a graph G on m edges with the property that every colouring of the edges of G with two colours yields a monochromatic copy of F. In 1983, Beck provided a beautiful argument that shows that $\^{r} $(Pn) is linear, solving a problem of Erdős. In this note, we provide another proof of this fact that actually gives a better bound, namely, $\^{r} $(Pn) < 137n for n sufficiently large.

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On-line Ramsey numbers for paths and stars

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.427 ◽

2008 ◽

Vol Vol. 10 no. 3 (Graph and Algorithms) ◽

Author(s):

J. A. Grytczuk ◽

H. A. Kierstead ◽

P. Prałat

Keyword(s):

Upper Bound ◽

Ramsey Number ◽

Ramsey Numbers ◽

Minimum Number ◽

International Audience ◽

On Line ◽

Monochromatic Copy

Graphs and Algorithms International audience We study on-line version of size-Ramsey numbers of graphs deﬁned via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a ﬁxed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number r(H) of the graph H. We determine exact values of r(H) for a few short paths and obtain a general upper bound r(Pn) ≤ 4n −7. We also study asymmetric version of this parameter when one of the target graphs is a star Sn with n edges. We prove that r(Sn, H) ≤ n*e(H) when H is any tree, cycle or clique

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A Note on Restricted Online Ramsey Numbers of Matchings

The Electronic Journal of Combinatorics ◽

10.37236/10025 ◽

2021 ◽

Vol 28 (3) ◽

Author(s):

Vojtěch Dvořák

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Short Note ◽

Ramsey Number ◽

Ramsey Numbers ◽

Minimum Number

Consider the following game between Builder and Painter. We take some families of graphs $\mathcal{G}_{1},\ldots,\mathcal{G}_t$ and an integer $n$ such that $n \geq R(\mathcal{G}_1,\ldots,\mathcal{G}_t)$. In each turn, Builder picks an edge of initially uncoloured $K_n$ and Painter colours that edge with some colour $i \in \left\{ 1,\ldots,t \right\}$ of her choice. The game ends when a graph $G_i$ in colour $i $ for some $G_i \in \mathcal{G}_i$ and some $i$ is created. The restricted online Ramsey number $\tilde{R}(\mathcal{G}_{1},\ldots,\mathcal{G}_t;n)$ is the minimum number of turns that Builder needs to guarantee the game to end. In a recent paper, Briggs and Cox studied the restricted online Ramsey numbers of matchings and determined a general upper bound for them. They proved that for $n=3r-1=R_2(r K_2)$ we have $\tilde{R}_{2}(r K_2;n) \leq n-1$ and asked whether this was tight. In this short note, we provide a general lower bound for these Ramsey numbers. As a corollary, we answer this question of Briggs and Cox, and confirm that for $n=3r-1$ we have $\tilde{R}_{2}(r K_2;n) = n-1$. We also show that for $n'=4r-2=R_3(r K_2)$ we have $\tilde{R}_{3}(r K_2;n') = 5r-4$.

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A short note on extreme points of certain polytopes

Special Matrices ◽

10.1515/spma-2020-0005 ◽

2020 ◽

Vol 8 (1) ◽

pp. 36-39

Author(s):

Lei Cao ◽

Ariana Hall ◽

Selcuk Koyuncu

Keyword(s):

Short Proof ◽

Convex Polytopes ◽

Short Note ◽

Convex Polytope ◽

Extreme Points ◽

Alternative Proof ◽

Stochastic Matrices ◽

Doubly Stochastic ◽

Birkhoff's Theorem ◽

Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

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

Mathematics ◽

10.3390/math9070764 ◽

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.

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A Study on f Number of Points for Number of Points with Inter-Distance of Less Than, or for Number of Points with Inter-Distance of or More to Necessarily Exist on Plane

10.31219/osf.io/vr697 ◽

2018 ◽

Author(s):

Benjamin Smith

Keyword(s):

Upper Bound ◽

Ramsey Number ◽

Minimum Value ◽

Special Case

We defined number of points with an inter-distance of β or more to necessarily exist on a plane. Furthermore, we aimed to reduce the range of this minimum value. We first showed that the upper bound of this value could be scaled by , and further reduced the constant that was multiplied. We compared the upper bound of and the Ramsey number in a special case and confirmed that was a better upper bound than except when were both small or trivial.

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