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

*Vince Grolmusz*

#### Abstract

*Alon, Kleitman, Lipton, Meshulam, Rabin*and

*Spencer*(Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph

**={F1,F2,…, Fd(q-1)+1}, where q is a prime-power, and d denotes the maximal degree of the hypergraph, there exists an**

*F***0⊂**

*F***, such that |⋃F∈**

*F***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(d2) 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.**

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