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

Paulo Henrique Cunha Gomes

TL;DR

The paper presents an explicit construction of a family $\mathcal{B}$ of $30$ blocks of size $6$ from $[60]$ such that every $6$-subset $S$ intersects some block in at least two points. This is achieved via a fully explicit partitioning of $[60]$ into ten base blocks, forming five pairs, splitting each base block into two triples, and generating 20 recombined blocks by pairing triples from each block pair. The result is a concrete, combinatorial covering-type object in the Johnson space $J(60,6)$ with practical relevance to covering designs. An explicit instance of the 30 blocks is provided, illustrating the construction and verification of the property. The work contributes to explicit design constructions and Johnson-space coverings by demonstrating a simple, modular method to achieve the stated intersection guarantee.

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.

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

TL;DR

The paper presents an explicit construction of a family of blocks of size from such that every -subset intersects some block in at least two points. This is achieved via a fully explicit partitioning of into ten base blocks, forming five pairs, splitting each base block into two triples, and generating 20 recombined blocks by pairing triples from each block pair. The result is a concrete, combinatorial covering-type object in the Johnson space with practical relevance to covering designs. An explicit instance of the 30 blocks is provided, illustrating the construction and verification of the property. The work contributes to explicit design constructions and Johnson-space coverings by demonstrating a simple, modular method to achieve the stated intersection guarantee.

Abstract

We exhibit an explicit family of subsets (``blocks'') of size of with the following property: for every -subset , there exists a block such that . The construction is fully explicit and the proof is purely combinatorial.
Paper Structure (5 sections, 1 theorem, 6 equations)

This paper contains 5 sections, 1 theorem, 6 equations.

Table of Contents

  1. Introduction
  2. Construction
  3. Main result
  4. An explicit example (the 30 blocks)
  5. Final remark

Key Result

Theorem 1

Let $\mathcal{B}$ be the family of $30$ blocks constructed above. For every $S\subset[60]$ with $|S|=6$, there exists $B\in\mathcal{B}$ such that $|S\cap B|\ge 2$. Equivalently: every $6$-subset $S$ contains a pair $\{x,y\}\subset S$ that is contained in at least one block $B\in\mathcal{B}$.

Theorems & Definitions (2)

  • Theorem 1
  • proof