A Sidon-type condition on set systems

Consider families of $k$-subsets (or blocks) on a ground set of size $v$. Recall that if all $t$-subsets occur with the same frequency $λ$, one obtains a $t$-design with index $λ$. On the other hand, if all $t$-subsets occur with different frequencies, such a family has been called (by Sarvate and others) a $t$-adesign. An elementary observation shows that such families always exist for $v > k \ge t$. Here, we study the smallest possible maximum frequency $μ=μ(t,k,v)$. The exact value of $μ$ is noted for $t=1$ and an upper bound (best possible up to a constant multiple) is obtained for $t=2$ using PBD closure. Weaker, yet still reasonable asymptotic bounds on $μ$ for higher $t$ follow from a probabilistic argument. Some connections are made with the famous Sidon problem of additive number theory.

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