In this paper we introduce the concept of minimum-weight edge-discriminators
in hypergraphs, and study its various properties. For a hypergraph H=(V,E), a function Ξ»:VβZ+βͺ{0} is said to be an {\it edge-discriminator} on H if
βvβEiββΞ»(v)>0, for all hyperedges EiββE, and
βvβEiββΞ»(v)ξ =βvβEjββΞ»(v), for every two
distinct hyperedges Eiβ,EjββE. An {\it optimal
edge-discriminator} on H, to be denoted by Ξ»Hβ, is
an edge-discriminator on H satisfying βvβVβΞ»Hβ(v)=minΞ»ββvβVβΞ»(v), where
the minimum is taken over all edge-discriminators on H. We prove
that any hypergraph H=(V,E), with β£Eβ£=n, satisfies βvβVβΞ»Hβ(v)β€n(n+1)/2,
and equality holds if and only if the elements of E are mutually
disjoint. For r-uniform hypergraphs H=(V,E), it
follows from results on Sidon sequences that βvβVβΞ»Hβ(v)β€β£Vβ£r+1+o(β£Vβ£r+1), and
the bound is attained up to a constant factor by the complete r-uniform
hypergraph. Next, we construct optimal edge-discriminators for some special
hypergraphs, which include paths, cycles, and complete r-partite hypergraphs.
Finally, we show that no optimal edge-discriminator on any hypergraph H=(V,E), with β£Eβ£=n(β₯3), satisfies
βvβVβΞ»Hβ(v)=n(n+1)/2β1, which, in turn,
raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure