Ramsey numbers avoiding properly colored cycles

Ramsey and 2-local Ramsey numbers for disjoint unions of cycles

The Ramsey number R(F) of a graph F without isolated vertices is the smallest positive integer n such that every red–blue coloring of Kn produces a subgraph isomorphic to F all of whose edges are colored the same. Let F be a set of graphs without isolated vertices. For a positive integer t, the vertex-disjoint Ramsey number VRt(F) is the smallest positive integer n such that every red–blue coloring of the complete graph Kn of order n results in at least t pairwise vertex-disjoint monochromatic graphs in F; while the edge-disjoint Ramsey number ERt(F) is the smallest positive integer n such that every red–blue coloring of Kn produces at least t pairwise edge-disjoint monochromatic graphs in F. If t=1 and F consists of a single graph F, then VR1(F)=ER1(F)=R(F) is the Ramsey number of the graph F. Thus, the concepts of vertex-disjoint and edge-disjoint Ramsey numbers provide a generalization of the standard Ramsey number. Upper and lower bounds for VRt(F) and ERt(F) are established for sets F of graphs without isolated vertices and the sharpness of these bounds is discussed. The primary goal of this paper is to investigate the values of VRt(F) and ERt(F) for sets F of graphs of size 2 or 3 without isolated vertices. The exact values of VRt(F) are determined for all such sets F and all integers t≥2. The exact values of ERt(F) of certain such sets F with prescribed conditions for all integers t≥2 are determined. For some special sets F of graphs of size 2 or 3 without isolated vertices, the exact values of ERt(F) are determined for 2≤t≤4. Additional results, problems, and conjectures are also presented dealing with these two Ramsey concepts for graphs in general.

The Ramsey number \(R_X(p,q)\) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey number is linear in X if there is a constant k such that \(R_{X}(p,q) \le k(p+q)\) for all p, q. In the present paper we conjecture that Ramsey number is linear in X if and only if the co-chromatic number is bounded in X and determine Ramsey numbers for several classes of graphs that verify the conjecture.

The Ramsey number is of vital importance in Ramsey's theorem. This paper proposed a novel methodology for constructing Ramsey graphs about R(3, 10), which uses Artificial Bee Colony optimization(ABC) to raise the lower bound of Ramsey number R(3, 10). The r(3, 10)-graph contains two limitations, that is, neither complete graphs of order 3 nor independent sets of order 10. To resolve these limitations, a special mathematical model is put in the paradigm to convert the problems into discrete optimization whose smaller minimizers are correspondent to bigger lower bound as approximation of inf R(3, 10). To demonstrate the potential of the proposed method, simulations are done to to minimize the amount of these two types of graphs. For the first time, four r(3, 9, 39) graphs with best approximation for inf R(3, 10) are reported in simulations to support the current lower bound for R(3, 10). The experiments' results show that the proposed paradigm for Ramsey number's calculation driven by ABC is a successful method with the advantages of high precision and robustness.

All Ramsey numbers for cycles in graphs

The Ramsey number is a foundational result in combinatorics. This article will introduce Ramsey number with the method of graph theory, and the Ramsey pricing theory is applied to the sales price and study of cross subsidy. Based on the status of our sales price and cross subsidy, Ramsey pricing methods theoretically guide adjustment thoughts of sales price and solve the practical problems in our life.

In previous work, the Ramsey numbers have been evaluated for all pairs of graphs with at most four points. In the present note, Ramsey numbers are tabulated for pairs F1, F2 of graphs where F1 has at most four points and F2 has exactly five points. Exact results are listed for almost all of these pairs.

On fan–wheel and tree–wheel Ramsey numbers

Given two graphs $G_1$ and $G_2$, the Ramsey number $R(G_1, G_2)$ is the smallest integer $N$ such that, for any graph $G$ of order $N$, either $G_1$ is a subgraph of $G$, or $G_2$ is a subgraph of the complement of $G$. Let $P_n$ denote a path of order $n$ and $\widehat{K}_m$ a kipas of order $m+1$, i.e., the graph obtained from a $P_m$ by adding one new vertex $v$ and edges from $v$ to all vertices of the $P_m$.We close the gap in existing knowledge on exact values of the Ramsey numbers $R(P_n,\widehat{K}_m)$ by determining the exact values for the remaining open cases.

We introduce an idea or a concept of restricted coexistence. By the restricted coexistence, the Ramsey number R(p,q) is defined equivalently as r(p-1,q), that is, R(p,q)=r(p-1,q), where r(p-1,q) is a least integer that has coexistence restricted to the parameters p-1,q with q ges p ges 2. From this, some basic properties about Ramsey numbers are obtained, for instance, R(p,q)>R(p-1,q+1), where p,q are integers with qgespges3, and so on.

Some Ramsey numbers for directed graphs

Given two graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that, for any graph G of order N, either G1 is a subgraph of G, or G2 is a subgraph of the complement of G. Let Cm denote a cycle of order m, K1,n a star of order n + 1 and Wn a wheel of order n + 1. Already back in the 1970s, exact values of the Ramsey numbers R(Cm,K1,n) have been determined for all m ≥ 2n and for all odd m ≤ 2n − 1, but for even m < 2n not many exact values are known. In this paper, we use a result of Bondy on pancyclicity to fill a considerable part of this gap. We show that R(Cm,K1,n) = 2n for even m with n < m < 2n, and that R(Cm,K1,n) = 2m−1 for even m with 3n/4+1 ≤ m ≤ n. In addition, we determine another six formerly unknown exact values of Ramsey numbers, namely R(C6,K1,n) for 7 ≤ n ≤ 11, and R(C6,W9).

Two remarks on the Burr–Erdős conjecture

Resolving a problem of Conlon, Fox, and R\"{o}dl, we construct a family of hypergraphs with arbitrarily large tower height separation between their $2$-colour and $q$-colour Ramsey numbers. The main lemma underlying this construction is a new variant of the Erd\H{o}s--Hajnal stepping-up lemma for a generalized Ramsey number $r_k(t;q,p)$, which we define as the smallest integer $n$ such that every $q$-colouring of the $k$-sets on $n$ vertices contains a set of $t$ vertices spanning fewer than $p$ colours. Our results provide the first tower-type lower bounds on these numbers.

For two given graphs $$G_1$$ and $$G_2$$ , the Ramsey number $$R(G_1,G_2)$$ is the smallest integer n such that for any graph G of order n, either G contains $$G_1$$ or its complement $${\overline{G}}$$ contains $$G_2$$ . Let $$P_n, S_n$$ and $$T_n$$ denote a path, a star and a tree of order n, respectively. A generalized wheel, denoted by $$W_{s,m}$$ , is the join of a complete graph $$K_s$$ and a cycle $$C_m$$ . In this paper, we show that $$R(T_n,W_{s,4})=(n-1)(s+1)+1$$ for $$n\ge 3,s\ge 2$$ and $$R(T_n,W_{s,5})=(n-1)(s+2)+1$$ for $$n\ge 3,s\ge 1$$ . These generalize some known results on Ramsey numbers for a tree versus a wheel.