Multicolour Bipartite Ramsey Number of Paths

Abstract

The k-colour bipartite Ramsey number of a bipartite graph H is the least integer N for which every k-edge-coloured complete bipartite graph K N,N contains a monochromatic copy of H. The study of bipartite Ramsey numbers was initiated over 40 years ago by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the 2-colour bipartite Ramsey number of paths. Recently the 3-colour Ramsey number of paths and (even) cycles, was essentially determined as well. Improving the results of DeBiasio, Gyárfás, Krueger, Ruszinkó, and Sárközy, in this paper we determine asymptotically the 4-colour bipartite Ramsey number of paths and cycles. We also provide new upper bounds on the k-colour bipartite Ramsey numbers of paths and cycles which are close to being tight.Over the years, many generalisations of Ramsey numbers have been considered (an excellent survey [3] by Conlon, Fox and Sudakov contains many examples); one natural example that we consider here

Monochromatic balanced components, matchings, and paths in multicolored complete bipartite graphs

“…For r = 2 an affirmative answer to Question 1.2 has been known in its strongest form for more than forty years: Gyárfás and Lehel [12] and independently Faudree and Schelp [7] proved that any 2-colored K n,n contains a monochromatic P n . For r = 3 an affirmative answer was recently provided by Bucić, Letzter and Sudakov [4] (In fact, the authors of this note independently proved the same result, but with a less elegant proof).…”

“…The significance of connected matchings is that with the connected matching-Regularity Lemma method established by Luczak [18], it is possible to transfer results on connected matchings to asymptotic results for paths and even cycles. For example, this method is used to transfer Theorem 1.3 to asymptotic results for even cycles and paths in [4]. For similar applications see for example [2], [8], [14], [16], [19].…”

“…It would be interesting to know whether we can additionally require that this large component be balanced; that is, is it true that in every r-coloring of K n,n there is a monochromatic component that meets both sides in at least n/r vertices?Over forty years ago, Gyárfás and Lehel [12] and independently Faudree and Schelp [7] proved that any 2-colored K n,n contains a monochromatic P n . Very recently, Bucić, Letzter and Sudakov [4] proved that every 3-colored K n,n contains a monochromatic connected matching (a matching whose edges are in the same connected component) of size ⌈n/3⌉. So the answer is strongly "yes" for 1 ≤ r ≤ 3.We provide a short proof of (a non-symmetric version of) the original question for 1 ≤ r ≤ 3; that is, every r-coloring of K m,n has a monochromatic component that meets each side in a 1/r proportion of its part size.…”

“…Over forty years ago, Gyárfás and Lehel [12] and independently Faudree and Schelp [7] proved that any 2-colored K n,n contains a monochromatic P n . Very recently, Bucić, Letzter and Sudakov [4] proved that every 3-colored K n,n contains a monochromatic connected matching (a matching whose edges are in the same connected component) of size ⌈n/3⌉. So the answer is strongly "yes" for 1 ≤ r ≤ 3.…”

Three-Color Bipartite Ramsey Number for Graphs with Small Bandwidth

“…For two colours, independently, Gyarfás and Lehel [14] and Faudree and Schelp [10] determined the exact bipartite Ramsey number for paths, showing that R bip pP n , P n q " n for odd n, and R bip pP n , P n q " n´1 for even n. In [35], Zhang and Sun proved that R bip pC 2n , C 4 q " n`1, and in [36] Zhang, Sun and Wu proved that R bip pC 2n , C 6 q " n`2 for n ě 4. The methods developed by Bucić, Letzter and Sudakov in [6] proves that R bip pC 2n , C 2m q "`1`op1q˘pn`mq for all positive n and m, determining asymptotically the bipartite Ramsey number of every pair of cycles. For complete bipartite graphs, the best known bounds for R bip pK n,n , K n,n q differ exponentially (see [7,17]).…”