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تسجيل مستخدم جديدOn the Ramsey number of the triangle and the cube
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The Ramsey number r(K_3,Q_n) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K_N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and ErdH{o}s conjectured that r(K_3,Q_n) = 2^{n+1} - 1 for every n in N, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K_3,Q_n) le 7000 cdot 2^n. Here we show that r(K_3,Q_n) = (1 + o(1)) 2^{n+1} as n to infty.
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The areas of Ramsey theory and random graphs have been closely linked ever since ErdH{o}s famous proof in 1947 that the diagonal Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was introduced as a model
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