Minimum blocking sets for families of partitions

Abstract

A $3$-partition of an $n$-element set $V$ is a triple of pairwise disjoint nonempty subsets $X,Y,Z$ such that $V=X\cup Y\cup Z$. We determine the minimum size $\varphi_3(n)$ of a set $\mathcal{E}$ of triples such that for every 3-partition $X,Y,Z$ of the set $\{1,\dots,n\}$, there is some $\{x,y,z\}\in \mathcal{E}$ with $x\in X$, $y\in Y$, and $z\in Z$. In particular, $$\varphi_3(n)=\left\lceil{\frac{n(n-2)}{3}}\right\rceil.$$ For $d>3$, one may define an analogous number $\varphi_d(n)$. We determine the order of magnitude of $\varphi_d(n)$, and prove the following upper and lower bounds, for $d>3$: $$\frac{2 n^{d-1}}{d!} -o(n^{d-1}) \leq \varphi_d(n) \leq \frac{0.86}{(d-1)!}n^{d-1}+o(n^{d-1}).$$