On Canonical Ramsey Numbers for Complete Graphs versus Paths

We consider a class of graphs on n vertices, called ( d, f)-arrangeable graphs. This class of graphs contains all graphs of bounded degree d, and all df-arrangeable graphs, a class introduced by Chen and Schelp in 1993. In 1992, a variation of the Regularity Lemma of Szemerédi was introduced by Eaton and Rödl. As an application of this lemma, we give a linear upper bound, c( d, f) n, for the Ramsey number of graphs in this class, where log 2 log 2 c( d, f) = 24 df 5. This improves the earlier result, given in 1983 by Chvátal et al. of a linear bound on the Ramsey number of graphs with bounded degree d, where the constant term was more that a tower of d 2's, and later extended by Chen and Schelp to include d-arrangeable graphs.

Two remarks on the Burr–Erdős conjecture

Edge-ordered Ramsey numbers

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.

Wheel and star-critical Ramsey numbers for quadrilateral

Finding Folkman Numbers via MAX CUT Problem

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.

Ramsey numbers for unions of some cycles

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.

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>For any fixed graphs $</span><span>G$ </span><span>and $</span><span>H$, </span><span>the Ramsey number $</span><span>R</span><span>(</span><span>G,H</span><span>)$ is the smallest positive integer $</span><span>n$ </span><span>such that for every graph $</span><span>F$ </span><span>on $</span><span>n$ </span><span>vertices must contain $</span><span>G$ </span><span>or the complement of $</span><span>F$ </span><span>contains $</span><span>H$. </span><span>The girth of graph $</span><span>G$ </span><span>is a length of the shortest cycle. A $</span><span>k$</span><span>-regular graph with the girth $</span><span>g$ </span><span>is called a $(</span><span>k,g</span><span>)$-graph. If the number of of vertices in $(</span><span>k,g</span><span>)$-graph is minimized then we call this graph a $(</span><span>k,g</span><span>)$-cage. In this paper, we derive the bounds of Ramsey number $</span><span>R</span><span>(</span><span>C_</span><span>4</span><span>,W_</span><span>n</span><span>)$ for some values of $</span><span>n$</span><span>. By modifying $(</span><span>k, </span><span>5)$-graphs, for $</span><span>k </span><span>= 7$ or $9$, we construct these corresponding $(</span><span>C_</span><span>4</span><span>,W_</span><span>n</span><span>)$-good graphs. </span></p></div></div></div>

Some Ramsey numbers for directed graphs

For a class of index coding problems with side information graph having the Ramsey number R(i, j) upper bounded by ci <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> j <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sup> , it is shown that the clique covering scheme approximates the broadcast rate within a multiplicative factor of O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a+b/a+b+1</sup> ), where n is the number of messages. Based on this result and known bounds on Ramsey numbers, it is demonstrated that the broadcast rate of planar graphs, line graphs, and fuzzy circular interval graphs can be approximated within a factor of (2n) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/3</sup> .

An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An N N -vertex graph is called C C -Ramsey if it has no homogeneous set of size C log ⁡ N C\log N . A theorem of Bukh and Sudakov, solving a conjecture of Erdős, Faudree, and Sós, shows that any C C -Ramsey N N -vertex graph contains an induced subgraph with Ω C ( N 1 / 2 ) \Omega _C(N^{1/2}) distinct degrees. We improve this to Ω C ( N 2 / 3 ) \Omega _C(N^{2/3}) , which is tight up to the constant factor. We also show that any N N -vertex graph with N &gt; ( k − 1 ) ( n − 1 ) N &gt; (k-1)(n-1) and n ≥ n 0 ( k ) = Ω ( k 9 ) n\geq n_0(k) = \Omega (k^9) either contains a homogeneous set of order n n or an induced subgraph with k k distinct degrees. The lower bound on N N here is sharp, as shown by an appropriate Turán graph, and confirms a conjecture of Narayanan and Tomon.

Relations between 2-local and 2-mean Ramsey numbers for graphs