The Ramsey number of books

Ramsey's Theorem is among the most well-known results in combinatorics. The theorem states that any two-edge-coloring of a sufficiently large complete graph contains a large monochromatic complete subgraph. Indeed, any two-edge-coloring of a complete graph with N=4t−o(t) vertices contains a monochromatic copy of Kt. On the other hand, a probabilistic argument yields that there exists a two-edge-coloring of a complete graph with N=2t/2+o(t) with no monochromatic copy of Kt. Much attention has been paid to improving these classical bounds but only improvements to lower order terms have been obtained so far. A natural question in this setting is to ask whether every two-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of Kt that can be extended in many ways to a monochromatic copy of Kt+1. Specifically, Erdős, Faudree, Rousseau and Schelp in the 1970's asked whether every two-edge-coloring of KN contains a monochromatic copy of Kt that can be extended in at least (1−ok(1))2−tN ways to a monochromatic copy of Kt+1. A random two-edge-coloring of KN witnesses that this would be best possible. While the intuition coming from random constructions can be misleading, for example, a famous construction by Thomason shows the existence of a two-edge-coloring of a complete graph with fewer monochromatic copies of Kt than a random two-edge-coloring, this paper confirms that the intuition coming from a random construction is correct in this case. In particular, the author answers this question of Erdős et al. in the affirmative. The question can be phrased in the language of Ramsey theory as a problem on determining the Ramsey number of book graphs. A book graph B(k)t is a graph obtained from Kt by adding k new vertices and joining each new vertex to all the vertices of Kt. The main result of the paper asserts that every two-edge-coloring of a complete graph with N=2kt+ok(t) vertices contains a monochromatic copy of B(k)t.

A Note on Edge-Colourings Avoiding Rainbow $K_4$ and Monochromatic $K_m$

10.37236/257 ◽

2009 ◽

Vol 16 (1) ◽

Author(s):

Veselin Jungić ◽

Tomáš Kaiser ◽

Daniel Král'

Keyword(s):

Lower Bound ◽

Complete Graph ◽

Upper Bound ◽

Projective Planes ◽

Ramsey Number ◽

Complete Subgraph ◽

Finite Projective Planes ◽

Edge Colourings ◽

Edge Colouring

We study the mixed Ramsey number $maxR(n,{K_m},{K_r})$, defined as the maximum number of colours in an edge-colouring of the complete graph $K_n$, such that $K_n$ has no monochromatic complete subgraph on $m$ vertices and no rainbow complete subgraph on $r$ vertices. Improving an upper bound of Axenovich and Iverson, we show that $maxR(n,{K_m},{K_4}) \leq n^{3/2}\sqrt{2m}$ for all $m\geq 3$. Further, we discuss a possible way to improve their lower bound on $maxR(n,{K_4},{K_4})$ based on incidence graphs of finite projective planes.

Degree Ramsey Numbers of Closed Blowups of Trees

10.37236/2526 ◽

2014 ◽

Vol 21 (2) ◽

Cited By ~ 2

Author(s):

Paul Horn ◽

Kevin G. Milans ◽

Vojtěch Rödl

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Edge Coloring ◽

Ramsey Number ◽

Ramsey Numbers ◽

Delta G ◽

The degree Ramsey number of a graph $G$, denoted $R_\Delta(G;s)$, is $\min\{\Delta(H)\colon\, H\stackrel{s}{\to} G\}$, where $H\stackrel{s}{\to} G$ means that every $s$-edge-coloring of $H$ contains a monochromatic copy of $G$. The closed $k$-blowup of a graph is obtained by replacing every vertex with a clique of size $k$ and every edge with a complete bipartite graph where both partite sets have size $k$. We prove that there is a function $f$ such that $R_\Delta(G;s) \le f(\Delta(G), s)$ when $G$ is a closed blowup of a tree.

Ramsey Numbers of Ordered Graphs

10.37236/7816 ◽

2020 ◽

Vol 27 (1) ◽

Cited By ~ 1

Author(s):

Martin Balko ◽

Josef Cibulka ◽

Karel Král ◽

Jan Kynčl

Keyword(s):

Chromatic Number ◽

Complete Graph ◽

Positive Answer ◽

Ramsey Number ◽

Ramsey Numbers ◽

Minimum Number ◽

Ordered Graphs ◽

Constant Interval ◽

Monochromatic Copy ◽

Special Classes

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

On Some Ramsey and Turán-Type Numbers for Paths and Cycles

10.37236/1081 ◽

2006 ◽

Vol 13 (1) ◽

Cited By ~ 6

Author(s):

Tomasz Dzido ◽

Marek Kubale ◽

Konrad Piwakowski

Keyword(s):

Complete Graph ◽

Ramsey Number ◽

Ramsey Numbers ◽

Multicolor Ramsey Number ◽

For given graphs $G_{1}, G_{2}, ... , G_{k}$, where $k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, ... , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph on $n$ vertices with $k$ colors, there is always a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. Let $P_k$ (resp. $C_k$) be the path (resp. cycle) on $k$ vertices. In the paper we show that $R(P_3,C_k,C_k)=R(C_k,C_k)=2k-1$ for odd $k$. In addition, we provide the exact values for Ramsey numbers $R(P_{4}, P_{4}, C_{k})=k+2$ and $R(P_{3}, P_{5}, C_{k})=k+1$.

On Colorings Avoiding a Rainbow Cycle and a Fixed Monochromatic Subgraph

10.37236/303 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 1

Author(s):

Maria Axenovich ◽

JiHyeok Choi

Keyword(s):

Complete Graph ◽

Edge Coloring ◽

Bipartite Graphs ◽

Fixed Number ◽

Let $H$ and $G$ be two graphs on fixed number of vertices. An edge coloring of a complete graph is called $(H,G)$-good if there is no monochromatic copy of $G$ and no rainbow (totally multicolored) copy of $H$ in this coloring. As shown by Jamison and West, an $(H,G)$-good coloring of an arbitrarily large complete graph exists unless either $G$ is a star or $H$ is a forest. The largest number of colors in an $(H,G)$-good coloring of $K_n$ is denoted $maxR(n, G,H)$. For graphs $H$ which can not be vertex-partitioned into at most two induced forests, $maxR(n, G,H)$ has been determined asymptotically. Determining $maxR(n; G, H)$ is challenging for other graphs $H$, in particular for bipartite graphs or even for cycles. This manuscript treats the case when $H$ is a cycle. The value of $maxR(n, G, C_k)$ is determined for all graphs $G$ whose edges do not induce a star.

Balanced Avoidance Games on Random Graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3454 ◽

2005 ◽

Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽

Author(s):

Martin Marciniszyn ◽

Dieter Mitsche ◽

Miloš Stojaković

Keyword(s):

Random Graphs ◽

Complete Graph ◽

Optimal Strategy ◽

Random Order ◽

The Other ◽

Threshold Function ◽

Graph Theoretic ◽

International Audience ◽

Balanced Coloring ◽

International audience We introduce and study balanced online graph avoidance games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with $n$ vertices are revealed two at a time in a random order. In each move, Painter immediately and irrevocably decides on a balanced coloring of the new edge pair: either the first edge is colored red and the second one blue or vice versa. His goal is to avoid a monochromatic copy of a given fixed graph $H$ in both colors for as long as possible. The game ends as soon as the first monochromatic copy of $H$ has appeared. We show that the duration of the game is determined by a threshold function $m_H = m_H(n)$. More precisely, Painter will asymptotically almost surely (a.a.s.) lose the game after $m = \omega (m_H)$ edge pairs in the process. On the other hand, there is an essentially optimal strategy, that is, if the game lasts for $m = o(m_H)$ moves, then Painter will a.a.s. successfully avoid monochromatic copies of H using this strategy. Our attempt is to determine the threshold function for certain graph-theoretic structures, e.g., cycles.

Three Color Ramsey Numbers for Graphs with at most 4 Vertices

10.37236/2160 ◽

2012 ◽

Vol 19 (4) ◽

Author(s):

Luis Boza ◽

Janusz Dybizbański ◽

Tomasz Dzido

Keyword(s):

Complete Graph ◽

Ramsey Number ◽

Free Graph ◽

Ramsey Numbers ◽

For given graphs $H_{1}, H_{2}, H_{3}$, the 3-color Ramsey number $R(H_{1},$ $H_{2}, H_{3})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with $3$ colors, then it always contains a monochromatic copy of $H_{i}$ colored with $i$, for some $1 \leq i \leq 3$.We study the bounds on 3-color Ramsey numbers $R(H_1,H_2,H_3)$, where $H_i$ is an isolate-free graph different from $K_2$ with at most four vertices, establishing that $R(P_4,C_4,K_4)=14$, $R(C_4,K_3,K_4-e)=17$, $R(C_4,K_3+e,K_4-e)=17$, $R(C_4,K_4-e,K_4-e)=19$, $28\le R(C_4,K_4-e,K_4)\le36$, $R(K_3,K_4-e,K_4)\le41$, $R(K_4-e,K_4-e,K_4)\le59$ and $R(K_4-e,K_4,K_4)\le113$. Also, we prove that $R(K_3+e,K_4-e,K_4-e)=R(K_3,K_4-e,K_4-e)$, $R(C_4,K_3+e,K_4)\le\max\{R(C_4,K_3,K_4),29\}\le32$, $R(K_3+e,K_4-e,K_4)\le\max\{R(K_3,K_4-e,K_4),33\}\le41$ and $R(K_3+e,K_4,K_4)\le\max\{R(K_3,K_4,K_4),2R(K_3,K_3,K_4)+2\}\le79$.This paper is an extension of the article by Arste, Klamroth, Mengersen [Utilitas Mathematica, 1996].

On Ramsey Numbers of Sparse Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548303005728 ◽

2003 ◽

Vol 12 (5-6) ◽

pp. 627-641 ◽

Cited By ~ 18

Author(s):

Alexander Kostochka ◽

B Sudakov

Keyword(s):

Complete Graph ◽

Ramsey Number ◽

Ramsey Numbers ◽

Sparse Graphs ◽

Minimum Integer ◽

Image Position ◽

The Ramsey number, , of a graph G is the minimum integer N such that, in every 2-colouring of the edges of the complete graph on N vertices, there is a monochromatic copy of G. In 1975, Burr and Erdős posed a problem on Ramsey numbers of d-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most d. They conjectured that for every d there exists a constant c(d) such that for any d-degenerate graph G of order n.In this paper we prove that for each such G. In fact, we show that, for every , sufficiently large n, and any graph H of order , either H or its complement contains a (d,n)-common graph, that is, a graph in which every set of d vertices has at least n common neighbours. It is easy to see that any (d,n)-common graph contains every d-degenerate graph G of order n. We further show that, for every constant C, there is an n and a graph H of order such that neither H nor its complement contains a -common graph.

New Lower Bound for Multicolor Ramsey Numbers for Even Cycles

10.37236/1980 ◽

2005 ◽

Vol 12 (1) ◽

Cited By ~ 1

Author(s):

Tomasz Dzido ◽

Andrzej Nowik ◽

Piotr Szuca

Keyword(s):

Lower Bound ◽

Complete Graph ◽

Finite Family ◽

Ramsey Number ◽

Ramsey Numbers ◽

Multicolor Ramsey Number ◽

For given finite family of graphs $G_{1}, G_{2}, \ldots , G_{k}, k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, \ldots , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph on $n$ vertices with $k$ colors then there is always a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. We give a lower bound for $k-$color Ramsey number $R(C_{m}, C_{m}, \ldots , C_{m})$, where $m \geq 4$ is even and $C_{m}$ is the cycle on $m$ vertices.

Ramsey Numbers for Theta Graphs

International Journal of Combinatorics ◽

10.1155/2011/649687 ◽

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.