In this paper we study multipartite Ramsey numbers for odd cycles. Our main result is the proof of a conjecture of Gyárfás, Sárközy and Schelp [12]. Precisely, let n ≥ 5 be an arbitrary positive odd integer; then in any two-coloring of the edges of the complete 5-partite graph K(n−1)/2,(n−1)/2,(n−1)/2,(n−1)/2,1 there is a monochromatic cycle of length n. keywords: cycles, Ramsey number, Regular...

In this thesis, we present new results which are concerned with the following four coloring problems in Ramsey Theory. i. The finite form of Brown’s Lemma. ii. Some 2-color Rado numbers. iii. Ramsey results involving Fibonacci sequences. iv. Coloring the odd-distance plane graph. A few of the results in this thesis are cited from earlier works of other authors. Several results are cited from pa...

Given a graph H and a positive integer n, Anti-Ramsey number AR(n,H) is the maximum number of colors in an edge-coloring of Kn that contains no polychromatic copy of H. The anti-Ramsey numbers were introduced in the 1970s by Erd” os, Simonovits, and Sós, who among other things, determined this function for cliques. In general, few exact values of AR(n,H) are known. Let us call a graph H doubly ...

A (finite) sequence of nonnegative integers is graphic if it is the degree sequence of some simple graph G. Given graphs G1 and G2, we consider the smallest integer n such that for every n-term graphic sequence π, there is some graph G with degree sequence π with G1 ⊆ G or with G2 ⊆ G. When the phrase “some graph” in the prior sentence is replaced with “all graphs” the smallest such integer n i...

Let L be a disjoint union of nontrivial paths. Such a graph we call a linear forest. We study the relation between the 2-local Ramsey number R2-loc(L) and the Ramsey number R(L), where L is a linear forest. L will be called an (n, j)-linear forest if L has n vertices and j maximal paths having an odd number of vertices. If L is an (n, j)-linear forest, then R2-loc(L) = (3n − j)/2 + dj/2e −

For simple graphs G1 and G2, the size Ramsey multipartite number mj(G1, G2) is defined as the smallest natural number s such that any arbitrary two coloring of the graph Kj×s using the colors red and blue, contains a red G1 or a blue G2 as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers mj(nK2, Cm) for j ≥ 2 and m ∈ {3, 4, 5, 6}.