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In \\cite{GriLiLu}, Griggs, Lu, and the author conjectured that if a family $\\F$ of subset of $[n]$ does not contain four distinct sets $A$, $B$, $C$ and $D$ forming a diamond, namely $A\\subset B\\cap C$ and $B\\cup C\\subset D$, then $\\hb_n(\\F)\\le 2+\\lfloor\\frac{n^2}{4}\\rfloor/(n^2-n)$. Moreover, the upped bound is achieved by three types of families.\n  In this paper, we prove the upper bound in the conjecture is asymptotically correct. 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In \\cite{GriLiLu}, Griggs, Lu, and the author conjectured that if a family $\\F$ of subset of $[n]$ does not contain four distinct sets $A$, $B$, $C$ and $D$ forming a diamond, namely $A\\subset B\\cap C$ and $B\\cup C\\subset D$, then $\\hb_n(\\F)\\le 2+\\lfloor\\frac{n^2}{4}\\rfloor/(n^2-n)$. Moreover, the upped bound is achieved by three types of families.\n  In this paper, we prove the upper bound in the conjecture is asymptotically correct. 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