Generalized ramsey theory VIII. The size ramsey number of small graphs

Generalized Ramsey theory for graphs XII: Bipartite Ramsey sets

Glasgow Mathematical Journal ◽

10.1017/s0017089500004444 ◽

1981 ◽

Vol 22 (1) ◽

pp. 31-41 ◽

Cited By ~ 4

Author(s):

Frank Harary ◽

Heiko Harborth ◽

Ingrid Mengersen

Keyword(s):

Ramsey Theory ◽

Ramsey Number

Following the notation in Faudree and Schelp [3], we write G → (F, H) to mean that every 2-coloring of E(G), the edge set of G, contains a green (the first color) F or a red (the second color) H. Then the Ramsey number r(F, H) of two graphs F and H with no isolated vertices has been defined as the minimum p such that Kp → (F, H).

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Generalized Ramsey Theory for Graphs V. the Ramsey Number of a Digraph

Bulletin of the London Mathematical Society ◽

10.1112/blms/6.2.175 ◽

1974 ◽

Vol 6 (2) ◽

pp. 175-182 ◽

Cited By ~ 9

Author(s):

Frank Harary ◽

Pavol Hell

Keyword(s):

Ramsey Theory ◽

Ramsey Number

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A linear upper bound in zero-sum Ramsey theory

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171294000864 ◽

1994 ◽

Vol 17 (3) ◽

pp. 609-612 ◽

Cited By ~ 3

Author(s):

Yair Caro

Keyword(s):

Upper Bound ◽

Ramsey Theory ◽

Ramsey Number ◽

Uniform Hypergraph ◽

Zero Sum ◽

Positive Integers

Letn,randkbe positive integers such thatk|(nr). There exists a constantc(k,r)such that for fixedkandrand for every groupAof orderkR(Knr,A)≤n+c(k,r),whereR(Knr,A)is the zero-sum Ramsey number introduced by Bialostocki and Dierker [1], andKnris the completer-uniform hypergraph onn-vertices.

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Trees with an On-Line Degree Ramsey Number of Four

The Electronic Journal of Combinatorics ◽

10.37236/660 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 3

Author(s):

David Rolnick

Keyword(s):

Maximum Degree ◽

Ramsey Theory ◽

Complete Classification ◽

Ramsey Number ◽

On Line ◽

Delta G ◽

Goal Graph ◽

Monochromatic Copy

On-line Ramsey theory studies a graph-building game between two players. The player called Builder builds edges one at a time, and the player called Painter paints each new edge red or blue after it is built. The graph constructed is called the background graph. Builder's goal is to cause the background graph to contain a monochromatic copy of a given goal graph, and Painter's goal is to prevent this. In the $S_k$-game variant of the typical game, the background graph is constrained to have maximum degree no greater than $k$. The on-line degree Ramsey number $\mathring{R}_{\Delta}(G)$ of a graph $G$ is the minimum $k$ such that Builder wins an $S_k$-game in which $G$ is the goal graph. Butterfield et al. previously determined all graphs $G$ satisfying $\mathring{R}_{\Delta}(G)\le 3$. We provide a complete classification of trees $T$ satisfying $\mathring{R}_{\Delta}(T)=4$.

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On-line Ramsey Theory for Bounded Degree Graphs

The Electronic Journal of Combinatorics ◽

10.37236/623 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 9

Author(s):

Jane Butterfield ◽

Tracy Grauman ◽

William B. Kinnersley ◽

Kevin G. Milans ◽

Christopher Stocker ◽

...

Keyword(s):

Ramsey Theory ◽

Upper Bounds ◽

Ramsey Number ◽

Bounded Degree ◽

Linear Forest ◽

On Line ◽

Bounded Degree Graphs ◽

Vertex Degrees ◽

Delta G ◽

Monochromatic Copy

When graph Ramsey theory is viewed as a game, "Painter" 2-colors the edges of a graph presented by "Builder". Builder wins if every coloring has a monochromatic copy of a fixed graph $G$. In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class ${\cal H}$. Builder wins the game $(G,{\cal H})$ if a monochromatic copy of $G$ can be forced. The on-line degree Ramsey number $\mathring {R}_\Delta(G)$ is the least $k$ such that Builder wins $(G,{\cal H})$ when ${\mathcal H}$ is the class of graphs with maximum degree at most $k$. Our results include: 1) $\mathring {R}_\Delta(G)\!\le\!3$ if and only if $G$ is a linear forest or each component lies inside $K_{1,3}$. 2) $\mathring {R}_\Delta(G)\ge \Delta(G)+t-1$, where $t=\max_{uv\in E(G)}\min\{d(u),d(v)\}$. 3) $\mathring {R}_\Delta(G)\le d_1+d_2-1$ for a tree $G$, where $d_1$ and $d_2$ are two largest vertex degrees. 4) $4\le \mathring {R}_\Delta(C_n)\le 5$, with $\mathring {R}_\Delta(C_n)=4$ except for finitely many odd values of $n$. 5) $\mathring {R}_\Delta(G)\le6$ when $\Delta(G)\le 2$. The lower bounds come from strategies for Painter that color edges red whenever the red graph remains in a specified class. The upper bounds use a result showing that Builder may assume that Painter plays "consistently".

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Tripartite and Quadpartite Size Ramsey Numbers for All Pairs of Connected Graphs on Four Vertices

Handbook of Research on Advanced Applications of Graph Theory in Modern Society - Advances in Computer and Electrical Engineering ◽

10.4018/978-1-5225-9380-5.ch010 ◽

2020 ◽

pp. 251-266

Author(s):

Chula J. Jayawardene

Keyword(s):

Graph Theory ◽

Upper Bound ◽

Ramsey Theory ◽

Premature Death ◽

Formal Logic ◽

Young Age ◽

Ramsey Number ◽

Ramsey Numbers ◽

Connected Graphs ◽

Area Of Study

A popular area of graph theory is based on a paper written in 1930 by F. P. Ramsey titled “On a Problem on Formal Logic.” A theorem which was proved in his paper triggered the study of modern Ramsey theory. However, his premature death at the young age of 26 hindered the development of this area of study at the initial stages. The balanced size multipartite Ramsey number mj (H,G) is defined as the smallest positive number s such that Kj×s→ (H,G). There are 36 pairs of (H, G), when H, G represent connected graphs on four vertices (as there are only 6 non-isomorphic connected graphs on four vertices). In this chapter, the authors find mj (H, G) exhaustively for all such pairs in the tripartite case j=3, and in the quadpartite case j=4, excluding the case m4 (K4,K4). In this case, the only known result is that m4 (K4,K4) is greater than or equal to 4, since no upper bound has been found as yet.

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Recent results in partition (Ramsey) theory for finite lattices

Discrete Mathematics ◽

10.1016/0012-365x(81)90207-7 ◽

1981 ◽

Vol 35 (1-3) ◽

pp. 185-198 ◽

Cited By ~ 8

Author(s):

Hans Jürgen Prömel ◽

Bernd Voigt

Keyword(s):

Ramsey Theory ◽

Finite Lattices

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A Note on On-Line Ramsey Numbers for Some Paths

Mathematics ◽

10.3390/math9070735 ◽

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.

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