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تسجيل مستخدم جديدThe Ramsey number of the clique and the hypercube
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The Ramsey number r(K_s,Q_n) is the smallest positive integer N such that every red-blue colouring of the edges of the complete graph K_N on N vertices contains either a red n-dimensional hypercube, or a blue clique on s vertices. Answering a question of Burr and ErdH{o}s from 1983, and improving on recent results of Conlon, Fox, Lee and Sudakov, and of the current authors, we show that r(K_s,Q_n) = (s-1) (2^n - 1) + 1 for every s in N and every sufficiently large n in N.
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Let $q_{min}(G)$ stand for the smallest eigenvalue of the signless Laplacian of a graph $G$ of order $n.$ This paper gives some results on the following extremal problem: How large can $q_minleft( Gright) $ be if $G$ is a graph of order $n,$ with n
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We determine the Ramsey number of a connected clique matching. That is, we show that if $G$ is a $2$-edge-coloured complete graph on $(r^2 - r - 1)n - r + 1$ vertices, then there is a monochromatic connected subgraph containing $n$ disjoint copies of
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There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus in 1965. It is useful in practice if similar results hold for hypergraphs. However the obvious generalization of Motzkin and Strau
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Given graphs $G$ and $H$ and a positive integer $q$ say that $G$ is $q$-Ramsey for $H$, denoted $Grightarrow (H)_q$, if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The size-Ramsey number $hat{r}(H)$ of a graph $H$ is
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