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For $r$-graphs $G$ and $F_1,\\dots,F_s$, we write $G\\to(F_1,\\dots,F_s)$ if every $s$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$-th color for some $1\\leq i\\leq s$. Let $\\mathcal{R}(F_1,\\dots,F_s)$ denote the family of all $r$-graphs $G$ with $G\\to(F_1,\\dots,F_s)$. When $F_1=\\dots=F_s=F$, we write $\\mathcal{R}(F;s)=\\mathcal{R}(F_1,\\dots,F_s)$.\n  In this paper, we investigate when $\\mathcal{R}(H;s)\\subseteq\\mathcal{R}(Q_1,\\dots,Q_t)$ holds, where $H=H^{(r)}(n,p)$ is a random $r$-graph and $Q_1,\\dots,Q_t$ are fixed $r$-graphs. 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