Book Ramsey numbers and quasi-randomness
A set of n triangles sharing a common edge is called a book with n pages and is denoted by Bn. It is known that the Ramsey number r ( B n) $ satisfies r(Bn) = ( 4+o ( 1 ) ) n.$ We show that every red-blue edge colouring of K⌊(4-ε)n⌋ with no monochromatic Bn exhibits quasi-random properties when $\varepsilon$ tends to 0. This implies that there is a constant $c>0$ such that for every red-blue edge colouring of K r(Bn) there is a monochromatic Bn whose vertices span at least ⌊cn2 ⌋ edges of the same colour as the book. As an application we find the Ramsey number for a class of graphs. © 2005 Cambridge University Press.
Combinatorics Probability and Computing
Nikiforov, V., Rousseau, C., & Schelp, R. (2005). Book Ramsey numbers and quasi-randomness. Combinatorics Probability and Computing, 14 (2022-05-06), 851-860. https://doi.org/10.1017/S0963548305006929