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The best general upper bound $O_p(n^{\\frac{2p+2}{2p-1}})$ is due to the celebrated work of Alon and Pudl\\'ak~[GAFA, 2003]. Our main contributions include:\n  (1) We prove Kusner's conjecture for every dimension $n\\ge 1$ when $2\\le p\\le 4$. 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