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\emphAlon, Kleitman, Lipton, Meshulam, Rabin and \emphSpencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph \textbf\textitF=\F_1,F_2,\ldots, F_d(q-1)+1\, where q is a prime-power, and d denotes the maximal degree of the hypergraph, there exists an \textbf\textitF_0⊂ \textbf\textitF, such that |\bigcup_F∈\textbf\textitF_0F| ≡ 0 (q). We give a direct, alternative proof for this theorem, and we also show that an explicit construction exists for a hypergraph of degree d and size Ω (d^2) which does not contain a non-empty sub-hypergraph with a union of size 0 modulo 6, consequently, the theorem does not generalize for non-prime-power moduli.

A Note on Set Systems with no Union of Cardinality 0 modulo m