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On of the most classical problems in Ramsey theory concerns finding R(s,t), which is the smallest number n such that every red-blue colouring of the edges of the complete graph on n vertices contains a complete red subgraph on s vertices, or a complete blue subgraph on t vertices. Motivated by a question of David Angell, we study a variant of Ramsey numbers where some edges are coloured both red and blue, or: purple. Specifically, we are interested in the largest number g = g(s,t,n), for some s, t and n < R(s,t), such that there exists a red-blue-purple colouring of the edges of K_n with g purple edges, without a red-purple copy of K_s and without a blue-purple copy of K_t. We determine g asymptotically for a large family of parameters, exhibiting strong dependencies with Ramsey-Turán numbers. Joint work with Thomas Lesgourgues and Nye Taylor. 

Ramsey with purple edges