A Sidon-type condition on set systems

TL;DR

This paper investigates the minimal maximum frequency in set systems where all t-subsets have distinct occurrence counts, providing exact results for t=1, bounds for t=2, and asymptotic bounds for higher t, with connections to Sidon problems.

Contribution

It introduces the concept of a t-adesign, studies the minimal maximum frequency, and derives bounds using combinatorial and probabilistic methods, linking to Sidon problems.

Findings

Exact value of μ for t=1.

Upper bounds for μ when t=2.

Asymptotic bounds for μ for t>2.

Abstract

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 $\lambda$, one obtains a $t$-design with index $\lambda$. 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 $\mu=\mu(t,k,v)$. The exact value of $\mu$ 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 $\mu$ for higher $t$ follow from a probabilistic argument. Some connections are made with the famous Sidon problem of additive number theory.