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 $\lambda$, 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.Comment: 6 page

Transversal designs and induced decompositions of graphs

We prove that for every complete multipartite graph $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as ${n\choose 2}-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into edge-disjoint induced subgraphs isomorphic to~$F$. This result identifies and structurally explains a gap between the growth rates $O(n)$ and $\Omega(n^{3/2})$ on the minimum number of non-edges in graphs admitting an induced $F$-decomposition

Mutually orthogonal latin squares with large holes

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \ge 8(t+1)^2 m$

Pairwise balanced designs covered by bounded flats

We prove that for any $K$ and $d$, there exist, for all sufficiently large admissible $v$, a pairwise balanced design PBD$(v,K)$ of dimension $d$ for which all $d$-point-generated flats are bounded by a constant independent of $v$. We also tighten a prior upper bound for $K = \{3,4,5\}$, in which case there are no divisibility restrictions on the number of points. One consequence of this latter result is the construction of latin squares `covered' by small subsquares