An explicit family of 30 blocks meeting every 6-set of [60] in at least two points

TL;DR

This paper presents an explicit combinatorial construction of 30 blocks of size 6 within a 60-element set, ensuring every 6-element subset intersects with at least two blocks, advancing combinatorial design theory.

Contribution

It introduces a fully explicit family of 30 blocks meeting a specific intersection property with all 6-subsets of a 60-element set.

Findings

Explicit construction of 30 blocks meeting the intersection property

Proof is purely combinatorial and fully explicit

Advances understanding of combinatorial block designs

Abstract

We exhibit an explicit family $\mathcal{B}$ of $30$ subsets (``blocks'') of size $6$ of $[60]=\{1,2,\dots,60\}$ with the following property: for every $6$-subset $S\subset[60]$, there exists a block $B\in\mathcal{B}$ such that $|S\cap B|\ge 2$. The construction is fully explicit and the proof is purely combinatorial.