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We show that the size-Ramsey number of the $\sqrt{n} \times \sqrt{n}$ grid graph is $O(n^{5/4})$, improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.
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On size Ramsey number of paths, trees, and circuits. I
J. Beck · 1983
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J. Beck · 1990
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Rolling backwards can move you forward: on embedding problems in sparse expanders
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