3 research outputs found
A Sidon-type condition on set systems
Consider families of -subsets (or blocks) on a ground set of size .
Recall that if all -subsets occur with the same frequency , one
obtains a -design with index . On the other hand, if all
-subsets occur with different frequencies, such a family has been called (by
Sarvate and others) a -adesign. An elementary observation shows that such
families always exist for . Here, we study the smallest possible
maximum frequency .
The exact value of is noted for and an upper bound (best possible
up to a constant multiple) is obtained for using PBD closure. Weaker, yet
still reasonable asymptotic bounds on for higher follow from a
probabilistic argument. Some connections are made with the famous Sidon problem
of additive number theory.Comment: 6 page