IL-GILLIS 2018 Problem 7

A uniform covering of the integers $1,2,...,n$ is a finite multiset of subsets of $\{1,2,...,n\}$ , so that each number lies in the same amount of sets from the covering. A covering may contain the same subset multiple times, it must contain at least one subset, and it may contain the empty subset. For example, $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})$ is a uniform covering of $1,2,3,4$ (every number occurs in two sets). The covering containing only the empty set is also uniform (every number occurs in zero sets).

Given two uniform coverings, we define a new uniform covering, their sum (denoted by $\oplus$ ), by adding the sets from both coverings. For example: $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})\oplus(\{1\},\{2\},\{3\},\{4\})=$ $(\{1\},\{1\},\{1\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\})$ A uniform covering is called non-composite if it's not a sum of two uniform coverings.

Prove that for any $n\geq1$ , there are only finitely many non-composite uniform coverings of $1,2,...,n$ .