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 which might potentially provide good lower bounds for the off-diagonal Ramsey numbers $R(3,k)$. In this model, edges of $K_n$ are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted $G_{n,triangle}$. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kims celebrated result that $R(3,k) = Theta big( k^2 / log k big)$. In this paper we improve the results of both Bohman and Kim, and follow the triangle-free process all the way to its asymptotic end. In particular, we shall prove that $$ebig( G_{n,triangle} big) ,=, left( frac{1}{2sqrt{2}} + o(1) right) n^{3/2} sqrt{log n },$$ with high probability as $n to infty$. We also obtain several pseudorandom properties of $G_{n,triangle}$, and use them to bound its independence number, which gives as an immediate corollary $$R(3,k) , ge , left( frac{1}{4} - o(1) right) frac{k^2}{log k}.$$ This significantly improves Kims lower bound, and is within a factor of $4 + o(1)$ of the best known upper bound, proved by Shearer over 25 years ago.
تحميل البحث