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

Abstract

Alon, Kleitman, Lipton, Meshulam, Rabin and Spencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph F ={F 1,F 2,…, F d(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 0 F| ≡ 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.