Some Ramsey numbers for directed graphs

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

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.

On fan–wheel and tree–wheel Ramsey numbers

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.

The Ramsey number r(G) of a graph G is the smallest integer n such that any 2 colouring of the edges of a clique on n vertices contains a monochromatic copy of G. Determining the Ramsey number of G is a central problem of Ramsey theory with long and illustrious history. Despite this there are precious few classes of graphs G for which the value of r(G) is known exactly. One such family consists of large vertex disjoint unions of a fixed graph H, we denote such a graph, consisting of n copies of H by nH. This classical result was proved by Burr, Erd˝os and Spencer in 1975, who showed r(nH) = (2jHj􀀀a(H))n+c, for some c = c(H), provided n is large enough. Since it did not follow from their arguments, Burr, Erd˝os and Spencer further asked to determine the number of copies we need to take in order to see this long term behaviour and the value of c. More than 30 years ago Burr gave a way of determining c(H), which only applies when the number of copies n is triple exponential in jHj. In this paper we give an essentially tight answer to this very old problem of Burr, Erd˝os and Spencer by showing that the long term behaviour occurs already when the number of copies is single exponential.

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.

Wheel and star-critical Ramsey numbers for quadrilateral

A D-decomposition of a directed graph G is a collection of arc-disjoint subgraphs of G, each isomorphic to D, such that every arc of G belongs to exactly one subgraph. The λ-fold spectrum problem for a directed graph D asks for the set of all integers v such that the λ-fold complete symmetric directed graph K*λKv* admits a D-decomposition. A five-cycle (pentagon) has 4 non-isomorphic orientations. The λ-fold spectrum problem has been solved for one of these oriented pentagons. In this paper, we provide a complete solution for each of the remaining three orientations, proving that the necessary and sufficient condition is 5∣λv(v−1) in all cases.

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.

An improved upper bound for Ramsey number N (3, 3, 3, 3; 2)

The planar Ramsey number for C4 and K5 is 13

For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest integer $N$ such that for any graph $G$ of order $N$, either $G$ contains $F$ or the complement of $G$ contains $H$. Let $F_l$ denote a fan of order $2l+1$, which is $l$ triangles sharing exactly one vertex, and $K_n$ a complete graph of order $n$. Surahmat et al. conjectured that $R(F_l,K_n)=2l(n-1)+1$ for $l\geq n\geq 5$. In this paper, we show that the conjecture is true for n=5.

The Ramsey numbers [formula omitted] and [formula omitted

The Ramsey Numbers of R(G,H) is the smallest integer k such that for any graph F of order k, either F contains G or the complement of F (or ) contains H. Let Sn denoted star of order n and Wm a wheel of order m + 1. In this paper, we show that R(S20, W10) = 43, and if k is even, R(S2k+1, W2k) = 5k – 1 for k ≥ 6.