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(i.e., with probability tending to 1 as $k \\to \\infty$) the maximum independent sets of $K_p(2k+1, k)$ are precisely the sets $\\{A\\in V(K(2k+1,k)): x\\in A\\}$ ($x\\in [2k+1]$).\n  We also complete the determination of the order of magnitude of the \"threshold\" for the above property for general $k$ and $n\\geq 2k+2$. 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