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.
Highlights
Ramsey theory refers to a large body of mathematical results, which roughly say that any sufficiently large structure is guaranteed to have a large well-organised substructure
A rare exception is a recent result of Jenssen and Skokan [11], who showed that the k-colour Ramsey number of an odd cycle Cn is exactly 2k−1(n − 1) + 1 for all sufficiently large n; interestingly, this does not hold for all k and n, see Day and Johnson [4]
For a path Pn, the k-colour Ramsey number is known to be at least (k − 1 + o(1))n, and at most (k − 1/2 + o(1))n; the same bounds hold for even cycles Cn
Summary
Ramsey theory refers to a large body of mathematical results, which roughly say that any sufficiently large structure is guaranteed to have a large well-organised substructure. The best known lower bound for the k-colour bipartite Ramsey number of a path or a cycle of length 2n is 5n for k = 4 and (2k − 4)n for k 5, while the best known upper bound is k(1 + 1 − 2/k + o(1))n (which is roughly (2k − 1 + o(1))n for large k) Both results are due to DeBiasio, Gyarfas, Krueger, Ruszinko and Sarkozy [5] who say that obtaining improvement to either of these bounds would be very interesting. As an immediate corollary (using the lower bound mentioned above, see Theorem 17), we determine, asymptotically, the 4-colour bipartite Ramsey number of a path or a cycle. The 5-colour bipartite Ramsey number of a cycle or path of order 2(n + 1) is larger than 6.5n
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