Minimum blocking sets for families of partitions

TL;DR

This paper determines the minimum size of a set of triples that intersect all 3-partitions of an n-element set, extending the concept to higher dimensions and providing bounds for the general case.

Contribution

It explicitly calculates the exact minimum for 3-partitions and establishes asymptotic bounds for higher dimensions, advancing understanding of blocking sets for partitions.

Findings

Exact value of 41;3(n) = 4;n(n-2)/3;

Asymptotic bounds for 41;d(n) when d > 3

Extension of the concept to d-dimensional 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 $d>3$: $$\frac{2 n^{d-1}}{d!} -o(n^{d-1}) \leq \varphi_d(n) \leq \frac{0.86}{(d-1)!}n^{d-1}+o(n^{d-1}).$$