A 910-block explicit construction guaranteeing a triple intersection with every $6$-subset of $[60]$

Abstract

We present a simple explicit family $\mathcal{B}$ of $910$ $6$-subsets of $[60]=\{1,\dots ,60\}$ such that every $6$-subset $S\subset [60]$ intersects at least one block $B\in \mathcal{B}$ in at least three elements, i.e.\ $|S\cap B|\ge 3$. Equivalently, $\mathcal{B}$ is a covering (dominating set) of the Johnson graph $J(60,6)$ with covering radius $3$ in the Johnson metric. The construction is purely combinatorial, based on a fixed split of $[60]$ into two halves, a pairing of each half, and a pigeonhole argument. We also record a crude counting lower bound and a straightforward generalization to $[2m]$ (with $m$ even).