Note on the three-coloured bipartite Ramsey numbers for paths

Abstract

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $\mathrm{br}(G_1,G_2,\dots ,G_k)$ is the least positive integer p so that any coloring of the edges of $K_{p,p}$ with k colors will result in a copy of $G_i$ in the ith color for some i. In this paper, we investigate the appearance of simpler monochromatic graphs such as paths under a 3-colouring of the edges of a bipartite graph. we obtain the exact value of $\mathrm{br}(P_3,P_3,P_n)$, and $\mathrm{br}(P_3,P_3,\dots ,P_n)$ for $n\ge 4r+2$, and $\mathrm{br}(P_4,P_4,P_n)$ for $n\ge 4$ by a new method of proof.

Keywords:

• Ramsey theory
• bipartite Ramsey number
• path

2010 AMS subject classifications:

• 05D10
• 05C80

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The funding was supported by the National Science Foundation of China [grant numbers 12061059, 11601254, 11551001] and the Qinghai Key Laboratory of Internet of Things Project [grant number 2017-ZJ-Y21].