On ramsey numbers for books

Abstract

For n = 1, 2, . . . , let 6, = K2+ K,,. We pose the problem of determining the Ramsey numbers r (&, B,) and demonstrate that in many cases critical colorings are available from known examples of strongly regular graphs.

Book Ramsey Numbers and Quasi-Randomness

“…Books have attracted considerable attention in Ramsey graph theory (see, e.g. [8], [4] and [6]). In particular, Erdős et al [3] proved that for every fixed k,…”

“…On the other hand, Rousseau and Sheehan in [8] proved that r (B n ) ≤ 4n + 2, and observed that the Paley graphs imply r (B n ) = 4n + 2 whenever 4n + 1 is a prime power. Hence, r (B n ) = (4 + o (1)) n. It is reasonable investigate the structure of red-blue edge colorings of K (4−ε)n without monochromatic B n when ε tends to 0.…”

Size Multipartite Ramsey Numbers for Small Paths Versus Stripes

“…. These bipartite Ramsey numbers have been explored extensively in the last decade (see [2,3,4,5,6] [1,10]) in the last decade. However, not much papers have been published in this area except for paths and cycles versus some small classes of graphs.…”

A note on Ramsey numbers for books

“…The book Ramsey number rðB m ; B n Þ is the smallest integer r such that either B m & G or B n & G for every graph G of order r. The study of Ramsey numbers for books was initiated in [7] and continued in [5]. The following results are known.…”