Extremal theory and bipartite graph-tree Ramsey numbers

AbstractFor a positive integer n and graph B, fB(n) is the least integer m such that any graph of order n and minimal degree m has a copy of B. It will be show that if B is a bipartite graph with parts of order k and l (k⩽l), then there exists a positive constant c, such that for any tree Tn of order n and for any j (0⩽j⩽(k-1)), the Ramsey number r(Tn,B)⩽n+c·(fB(n))j/(k-1) if Δ(Tn)⩽(n/(k-j-1))-(j+2)·fB(n). In particular, this implies r(Tn, B) is bounde d above by n+o(n) for any tree Tn (since fB(n)=o(n) when B is a Bipartite graph), and by n+O(1) if the tree Tn has no vertex of large degree. For special classes of bipartite graphs, such as even cycles, sharper bounds will be proved along with examples demonstrating their sharpness. Also, applications of this to the determination of Ramsey number for arbitrary graphs and trees will be discussed

Ramsey Goodness and Beyond

In a seminal paper from 1983, Burr and Erdos 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 Szemeredi regularity lemma, embedding of sparse graphs, Turan type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving all but one of the Burr-Erdos problems.Comment: A new reference is adde

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

On Ramsey Numbers Involving Starlike Multipartite Graphs

https://digitalcommons.memphis.edu/speccoll-faudreerj/1104/thumbnail.jp