## DIMACS TR: 2000-11

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

### Author: Vince Grolmusz

**
ABSTRACT
**

Alon, Kleitman, Lipton, Meshulam, Rabin and Spencer (Graphs. Combin. 7
(1991),
no. 2, 97-99) proved, that for any hypergraph ${\cal
F}=\{F_1,F_2,\ldots, F_{d(q-1)+1}\}$, where $q$
is a prime-power, and $d$ denotes the maximal degree of the hypergraph,
there exists an
${\cal F}_0\subset {\cal F}$, such that $|\bigcup_{F\in{\cal
F}_0}F|\equiv 0\pmod{q}$. We give a direct, alternative
proof for this theorem, and we also give an explicit construction of a
hypergraph of degree
$d$ and size $\Omega(d^2)$ which does not contain a non-empty
sub-hypergraph with a union of size 0 modulo 6.

Paper Available at:
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