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

Discrete Mathematics & Theoretical Computer Science, Vol 6, No 1 (2003)

DMTCS vol 6 no 1 (2003), pp. 41-44

Discrete Mathematics & Theoretical Computer Science

Volume 6 n° 1 (2003), pp. 41-44

author:	Vince Grolmusz
title:	A Note on Set Systems with no Union of Cardinality 0 Modulo m
keywords:	hypergraphs, composite modulus, explicit constructions
abstract:	Alon, Kleitman, Lipton, Meshulam, Rabin and Spencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph F={F₁,F₂,…, F_d(q-1)+1}, where q is a prime-power, and d denotes the maximal degree of the hypergraph, there exists an F₀⊂ F, such that \|∪_F∈F₀F\| ≡ 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. If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files.
reference:	Vince Grolmusz (2003), A Note on Set Systems with no Union of Cardinality 0 Modulo m, Discrete Mathematics and Theoretical Computer Science 6, pp. 41-44
bibtex:	For a corresponding BibTeX entry, please consider our BibTeX-file.
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