The multicolour size Ramsey number of a path
Csongor Beke, Anqi Li, Julian Sahasrabudhe

TL;DR
This paper determines the multicolour size Ramsey number of paths, establishing tight bounds for fixed numbers of colours and path lengths, and improves the lower bound for these numbers.
Contribution
It provides the first tight bounds for the multicolour size Ramsey number of paths and enhances the lower bound for these numbers.
Findings
For fixed r ≥ 2 and k ≥ 100 log r, the size Ramsey number is Θ((r^2 log r) k).
Established tight bounds for the multicolour size Ramsey number of paths.
Improved the lower bound on the size Ramsey number of paths.
Abstract
In this paper, we determine the -colour size Ramsey number of the path , up to constants. In particular, for every fixed and , we have \[ \widehat{R}_r(P_k)=\Theta((r^2 \log r) \, k).\] Perhaps surprisingly, we do this by improving the lower bound on .
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