Ramsey with purple edges
Thomas Lesgourgues, Anita Liebenau, Nye Taylor
TL;DR
This work introduces and analyzes a red/blue/purple variant of Ramsey theory, focusing on the maximal number of purple edges $g(n;s,t)$ in an $(s,t)$-free colouring of the complete graph $K_n$ with $n<R(s,t)$. The authors establish a tight connection between $g(n;s,t)$ and Ramsey–Turán numbers ${\bf RT}_s(n,t)$ via blow-up constructions and extremal graphs, yielding asymptotics for broad parameter families and highlighting the triangle case as a central theme. In particular, for triangles with linear independence number ($t=cn$) they prove $g(n;3,cn)=(1-o(1)){\bf RT}_3(n,cn)$ for most $c$, with precise behavior tied to Andrásfai graphs in certain subranges. For sublinear $t$, the triangle case exhibits a distinct regime where $g(n;3,t)$ can be a positive fraction of $nt$, with lower bounds obtained through the triangle-free process. The paper also extends the analysis to general clique sizes, providing both explicit blow-up-based constructions and probabilistic lower bounds, and includes computational results that illuminate small-case behaviour and guide future questions. Overall, the work significantly advances understanding of how a third colour (purple) interacts with classical Ramsey–Turán phenomena and opens multiple avenues for further extremal and computational exploration.
Abstract
Motivated by a question of Angell, we investigate a variant of Ramsey numbers where some edges are coloured simultaneously red and blue, which we call purple. Specifically, we are interested in the largest number $g=g(n;s,t)$, for some $s$ and $t$ and $n<R(s,t)$, such that there exists a red/blue/purple colouring of $K_n$ with $g$ purple edges, with no red/purple copy of $K_s$ nor blue/purple copy of $K_t$. We determine $g$ asymptotically for a large family of parameters, exhibiting strong dependencies with Ramsey-Turán numbers.