Perfect partitions of a random set of integers

Abstract Let $X_1,\ldots, X_n$ be independent integers distributed uniformly on [ M ], $M\ge 2$ . A partition S of [ n ] into $\nu$ non-empty subsets $S_1,\ldots, S_{\nu}$ is called perfect if all $\nu$ values $\sum_{j\in S_{\alpha}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $\nu$ . In 2001, for $\nu=2$ , Christian Borgs, Jennifer Chayes, and the author proved that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $\kappa\;:\!=\; \lim {{n}/{\log M}}>{{1}/{\log 2}}$ , and that with high probability no perfect partition exists if $\kappa