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

TL;DR

This paper introduces an explicit combinatorial construction of 920 subsets of size 6 from a 60-element set, ensuring every 6-subset intersects with at least one block in three elements, useful for combinatorial design applications.

Contribution

The paper provides a novel explicit combinatorial construction guaranteeing triple intersections with all 6-subsets, advancing design theory methods.

Findings

Constructed a 920-block family with guaranteed triple intersections.

Used a partition of the set into pairs and pigeonhole principle.

Discussed how different partitions influence intersection properties.

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.