Conjecture on the exceptional thresholds for exponentially many perfect partitions
Let ν≥3\nu\ge 3ν≥3, let M=M(n)→∞M=M(n)\to\inftyM=M(n)→∞ with M=exp(O(n))M=\exp(O(n))M=exp(O(n)), and let X1,…,XnX_1,\dots,X_nX1,…,Xn be independent random variables uniformly distributed on [M]:={1,…,M}[M]:=\{1,\dots,M\}[M]:={1,…,M}. An ordered partiti…