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تسجيل مستخدم جديدThe $chi$-Ramsey problem for triangle-free graphs
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In 1967, ErdH{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of ErdH{o}s and Hajnal together with Shearers classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 sqrt{2} + o(1)) sqrt{n/log n}$. We improve this bound by a factor $sqrt{2}$, as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot.
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Given a class $mathcal{C}$ of graphs and a fixed graph $H$, the online Ramsey game for $H$ on $mathcal C$ is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each turn Builde
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For all $nge 9$, we show that the only triangle-free graphs on $n$ vertices maximizing the number $5$-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by ErdH{o}s, and extends results by Grzesik and Hatami, Hladky, Kr{
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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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