Improved bounds on the multicolor Ramsey numbers of paths and even cycles

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Multicolor Ramsey numbers for paths and cycles

For given graphs G₁,G₂,...,Gₖ, k ≥ 2, the multicolor Ramsey number R(G₁,G₂,...,Gₖ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors, then it is always a monochromatic copy of some $G_i$, for 1 ≤ i ≤ k. We give a lower bound for k-color Ramsey number R(Cₘ,Cₘ,...,Cₘ), where m ≥ 8 is even and Cₘ is the cycle on m vertices. In addition, we provide exact values for Ramsey numbers R(P₃,Cₘ,Cₚ), where P₃ is the path on 3 vertices, and several values for R(Pₗ,Pₘ,Cₚ), where l,m,p ≥ 2. In this paper we present new results in this field as well as some interesting conjectures