Cecil C. Rousseau

THE SIZE RAMSEY NUMBER

Let denote the class of all graphsG which satisfyG→(G1,G2). As a way of measuring minimality for members of, we define thesize Ramsey number ř(G1,G2) by.We then investigate various questions concerned with the asymptotic behaviour ofř.

On cycle—Complete graph ramsey numbers

A new upper bound is given for the cycle-complete graph Ramsey number r(C,,,, K„), the smallest order for a graph which forces it to contain either a cycle of order m or a set of ri independent vertices

On ramsey numbers for books

For n = 1, 2, …, let Bn = K2 + Kn. We pose the problem of determining the Ramsey numbers r(Bm, Bn) and demonstrate that in many cases critical colorings are avialable from known examples of strongly regular graphs.

Ramsey goodness and beyond

In a seminal paper from 1983, Burr and Erdős started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving five — all but one — of the Burr-Erdős problems.

On Book-Complete Graph Ramsey Numbers

It is shown that a graph of orderNand average degreedthat does not contain the bookBm=K1+K1, mas a subgraph has independence number at leastNf(d), wheref(x)~(logx/x)(x?∞). From this result we find that the book-complete graph Ramsey number satisfiesr(Bm, Kn)?mn2/log(n/e). It is also shown that for every treeTmwithmedges,r(K1+Tm,Kn)?(2m?1)n2/log(n/e).

Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

Abstract The Ramsey number r(H,Kn) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K 2,m ,K n )⩽(m−1+ o (1))(n/ log n) 2 and r(C 2m ,K n )⩽c(n/ log n) m/(m−1) for m fixed and n→∞. Also r(K 2,n ,K n )=Θ(n 3 / log 2 n) and r(C 5 ,K n )⩽cn 3/2 / log n .

Some Complete Bipartite Graph - Tree Ramsey Numbers

We investigate r(Ka,a,, T) for a = 2 and a = 3, where T is an arbitrary tree of order n. For a = 2, this Ramsey number is completely determined by r(K2,2, K1,m) where m = Δ(T). For a = 3, we do not find such an “exact” result, but we do show that r(K3,3, T)

Generalized Ramsey theory for multiple colors

In this paper, we study the generalized Ramsey number r(G1,…, Gk) where the graphs G1,…, Gk consist of complete graphs, complete bipartite graphs, paths, and cycles. Our main theorem gives the Ramsey number for the case where G2,…, Gk are fixed and G1 ⋍ Cn or Pn with n sufficiently large. If among G2,…, Gk there are both complete graphs and odd cycles, the main theorem requires an additional hypothesis concerning the size of the odd cycles relative to their number. If among G2,…, Gk there are odd cycles but no complete graphs, then no additional hypothesis is necessary and complete results can be expressed in terms of a new type of Ramsey number which is introduced in this paper. For k = 3 and k = 4 we determine all necessary values of the new Ramsey number and so obtain, in particular, explicit and complete results for the cycle Ramsey numbers r(Cn, Cl, Ck) and r(Cn, Cl, Ck, Cm) when n is large.

Asymptotic Upper Bounds for Ramsey Functions

Abstract. We show that for any graph G with N vertices and average degree d, if the average degree of any neighborhood induced subgraph is at most a, then the independence number of G is at least Nfa+1(d), where fa+1(d)=∫01(((1−t)1/(a+1))/(a+1+(d−a−1)t))dt. Based on this result, we prove that for any fixed k and l, there holds r(Kk+l,Kn)≤ (l+o(1))nk/(logn)k−1. In particular, r(Kk, Kn)≤(1+o(1))nk−1/(log n)k−2.

A local density condition for triangles

Let G be a graph on n vertices and let ? and β be real numbers, 0 < ?, β < 1. Further, let G satisfy the condition that each ??n? subset of its vertex set spans at least βn2 edges. The following question is considered. For a fixed ? what is the smallest value of β such that G contains a triangle?