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

TL;DR

This paper introduces a simple explicit construction of 910 six-element subsets of a 60-element set, ensuring every six-element subset intersects at least one block in three elements, with applications in combinatorics and Johnson graph coverings.

Contribution

The paper provides a novel explicit combinatorial construction of a covering set for the Johnson graph J(60,6) with a specific intersection property, extending to general even-sized sets.

Findings

Constructed a 910-block family covering all 6-subsets with 3-element intersections.

Proved the covering set is minimal or near-minimal based on a crude lower bound.

Extended the construction to larger even-sized sets, generalizing the approach.

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).