A 920-block explicit construction guaranteeing a triple intersection with every 6-subset of [60]

Abstract

We present an explicit family $\mathcal{B}$ of $920$ subsets of size $6$ of $[60]=\{1,\dots ,60\}$ with the property that every $6$-subset $S\subset [60]$ intersects at least one block $B\in \mathcal{B}$ in at least three elements, i.e.\ $|S\cap B|\ge 3$. The construction is purely combinatorial, based on a partition of the ground set into pairs and a pigeonhole argument. We also record a simple counting lower bound and discuss how different partitions of the ten base blocks affect the emergence of triple intersections.