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تسجيل مستخدم جديدThe size-Ramsey number of 3-uniform tight paths
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Given a hypergraph $H$, the size-Ramsey number $hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a monochromatic copy of $H$. We prove that the size-Ramsey number of the $3$-uniform tight path on $n$ vertices $P^{(3)}_n$ is linear in $n$, i.e., $hat{r}_2(P^{(3)}_n) = O(n)$. This answers a question by Dudek, Fleur, Mubayi, and Rodl for $3$-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved $hat{r}_2(P^{(3)}_n) = O(n^{3/2} log^{3/2} n)$.
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An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V(G)$. In on-line Ramsey theory, Builder iteratively presents edge
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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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